Elementary Functions - 4.24 Inverse Trigonometric Functions: Further Properties

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
4.24.E2 arccos ⁡ z = ( 2 ⁢ ( 1 - z ) ) 1 / 2 ⁢ ( 1 + ∑ n = 1 ∞ 1 ⋅ 3 ⋅ 5 ⁢ ⋯ ⁢ ( 2 ⁢ n - 1 ) 2 2 ⁢ n ⁢ ( 2 ⁢ n + 1 ) ⁢ n ! ⁢ ( 1 - z ) n ) 𝑧 superscript 2 1 𝑧 1 2 1 superscript subscript 𝑛 1 ⋅ 1 3 5 ⋯ 2 𝑛 1 superscript 2 2 𝑛 2 𝑛 1 𝑛 superscript 1 𝑧 𝑛 {\displaystyle{\displaystyle\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+% \sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}% \right)}}
\acos@@{z} = (2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right)		 | 1 - z | ≤ 2 1 𝑧 2 {\displaystyle{\displaystyle|1-z|\leq 2}} 
arccos(z) = (2*(1 - z))^(1/2)*(1 + sum((1 * 3 * 5*(2*n - 1))/((2)^(2*n)*(2*n + 1)*factorial(n))*(1 - z)^(n), n = 1..infinity))

ArcCos[z] == (2*(1 - z))^(1/2)*(1 + Sum[Divide[1 * 3 * 5*(2*n - 1),(2)^(2*n)*(2*n + 1)*(n)!]*(1 - z)^(n), {n, 1, Infinity}, GenerateConditions->None])
Failure Failure
Failed [7 / 7]
Result: .3065228369+.5552108774*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}


Result: -3.012742443+4.300365362*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}


Result: .3498215011-1.819822265*I
Test Values: {z = 1/2-1/2*I*3^(1/2)}


Result: -5.876013992-3.037981862*I
Test Values: {z = -1/2*3^(1/2)-1/2*I}

... skip entries to safe data
Failed [7 / 7]
Result: Complex[0.3065228364484756, 0.5552108781095243]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-3.0127424460165777, 4.300365361528893]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
4.24.E6 x 2 - y 2 = - 1 2 superscript 𝑥 2 superscript 𝑦 2 1 2 {\displaystyle{\displaystyle x^{2}-y^{2}=-\tfrac{1}{2}}}
x^{2}-y^{2} = -\tfrac{1}{2}		 

(x)^(2)- (y)^(2) = -(1)/(2)
 
(x)^(2)- (y)^(2) == -Divide[1,2]
 Skipped - no semantic math Skipped - no semantic math - -
4.24.E7 d d z ⁡ arcsin ⁡ z = ( 1 - z 2 ) - 1 / 2 derivative 𝑧 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsin% }z=(1-z^{2})^{-1/2}}}
\deriv{}{z}\asin@@{z} = (1-z^{2})^{-1/2}		 

diff(arcsin(z), z) = (1 - (z)^(2))^(- 1/2)

D[ArcSin[z], z] == (1 - (z)^(2))^(- 1/2)
Successful Successful - Successful [Tested: 7]
4.24.E8 d d z ⁡ arccos ⁡ z = - ( 1 - z 2 ) - 1 / 2 derivative 𝑧 𝑧 superscript 1 superscript 𝑧 2 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccos% }z=-(1-z^{2})^{-1/2}}}
\deriv{}{z}\acos@@{z} = -(1-z^{2})^{-1/2}		 

diff(arccos(z), z) = -(1 - (z)^(2))^(- 1/2)

D[ArcCos[z], z] == -(1 - (z)^(2))^(- 1/2)
Successful Successful - Successful [Tested: 7]
4.24.E9 d d z ⁡ arctan ⁡ z = 1 1 + z 2 derivative 𝑧 𝑧 1 1 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arctan% }z=\frac{1}{1+z^{2}}}}
\deriv{}{z}\atan@@{z} = \frac{1}{1+z^{2}}		 

diff(arctan(z), z) = (1)/(1 + (z)^(2))

D[ArcTan[z], z] == Divide[1,1 + (z)^(2)]
Successful Successful - Successful [Tested: 7]
4.24.E10 d d z ⁡ arccsc ⁡ z = - 1 z ⁢ ( z 2 - 1 ) 1 / 2 derivative 𝑧 𝑧 1 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc% }z=-\frac{1}{z(z^{2}-1)^{1/2}}}}
\deriv{}{z}\acsc@@{z} = -\frac{1}{z(z^{2}-1)^{1/2}}		 

diff(arccsc(z), z) = -(1)/(z*((z)^(2)- 1)^(1/2))

D[ArcCsc[z], z] == -Divide[1,z*((z)^(2)- 1)^(1/2)]
Failure Failure
Failed [2 / 7]
Result: 1.074569932-1.074569932*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2}


Result: -.6696152420e-9+2.000000000*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}

Successful [Tested: 1]
4.24.E10 d d z ⁡ arccsc ⁡ z = + 1 z ⁢ ( z 2 - 1 ) 1 / 2 derivative 𝑧 𝑧 1 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccsc% }z=+\frac{1}{z(z^{2}-1)^{1/2}}}}
\deriv{}{z}\acsc@@{z} = +\frac{1}{z(z^{2}-1)^{1/2}}		 

diff(arccsc(z), z) = +(1)/(z*((z)^(2)- 1)^(1/2))

D[ArcCsc[z], z] == +Divide[1,z*((z)^(2)- 1)^(1/2)]
Failure Failure
Failed [5 / 7]
Result: -.6696152420e-9+2.000000000*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2}


Result: 1.074569932-1.074569932*I
Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2}


Result: -1.192569588
Test Values: {z = 1.5, z = 1/2}


Result: 4.618802153*I
Test Values: {z = .5, z = 1/2}

... skip entries to safe data
Failed [1 / 1]
Result: Complex[0.0, 4.618802153517007]
Test Values: {Rule[z, Rational[1, 2]]}

4.24.E11 d d z ⁡ arcsec ⁡ z = + 1 z ⁢ ( z 2 - 1 ) 1 / 2 derivative 𝑧 𝑧 1 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec% }z=+\frac{1}{z(z^{2}-1)^{1/2}}}}
\deriv{}{z}\asec@@{z} = +\frac{1}{z(z^{2}-1)^{1/2}}		 

diff(arcsec(z), z) = +(1)/(z*((z)^(2)- 1)^(1/2))

D[ArcSec[z], z] == +Divide[1,z*((z)^(2)- 1)^(1/2)]
Failure Failure
Failed [2 / 7]
Result: -1.074569932+1.074569932*I
Test Values: {z = -1/2+1/2*I*3^(1/2), z = 1/2}


Result: .6696152420e-9-2.000000000*I
Test Values: {z = -1/2*3^(1/2)-1/2*I, z = 1/2}

Successful [Tested: 1]
4.24.E11 d d z ⁡ arcsec ⁡ z = - 1 z ⁢ ( z 2 - 1 ) 1 / 2 derivative 𝑧 𝑧 1 𝑧 superscript superscript 𝑧 2 1 1 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arcsec% }z=-\frac{1}{z(z^{2}-1)^{1/2}}}}
\deriv{}{z}\asec@@{z} = -\frac{1}{z(z^{2}-1)^{1/2}}		 

diff(arcsec(z), z) = -(1)/(z*((z)^(2)- 1)^(1/2))

D[ArcSec[z], z] == -Divide[1,z*((z)^(2)- 1)^(1/2)]
Failure Failure
Failed [5 / 7]
Result: .6696152420e-9-2.000000000*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, z = 1/2}


Result: -1.074569932+1.074569932*I
Test Values: {z = 1/2-1/2*I*3^(1/2), z = 1/2}


Result: 1.192569588
Test Values: {z = 1.5, z = 1/2}


Result: -4.618802153*I
Test Values: {z = .5, z = 1/2}

... skip entries to safe data
Failed [1 / 1]
Result: Complex[0.0, -4.618802153517007]
Test Values: {Rule[z, Rational[1, 2]]}

4.24.E12 d d z ⁡ arccot ⁡ z = - 1 1 + z 2 derivative 𝑧 𝑧 1 1 superscript 𝑧 2 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\operatorname{arccot% }z=-\frac{1}{1+z^{2}}}}
\deriv{}{z}\acot@@{z} = -\frac{1}{1+z^{2}}		 

diff(arccot(z), z) = -(1)/(1 + (z)^(2))

D[ArcCot[z], z] == -Divide[1,1 + (z)^(2)]
Successful Successful - Successful [Tested: 7]
4.24.E13 Arcsin ⁡ u + Arcsin ⁡ v = Arcsin ⁡ ( u ⁢ ( 1 - v 2 ) 1 / 2 + v ⁢ ( 1 - u 2 ) 1 / 2 ) multivalued-inverse-sine 𝑢 multivalued-inverse-sine 𝑣 multivalued-inverse-sine 𝑢 superscript 1 superscript 𝑣 2 1 2 𝑣 superscript 1 superscript 𝑢 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}u+\operatorname{Arcsin}v=% \operatorname{Arcsin}\left(u(1-v^{2})^{1/2}+v(1-u^{2})^{1/2}\right)}}
\Asin@@{u}+\Asin@@{v} = \Asin@{u(1-v^{2})^{1/2}+ v(1-u^{2})^{1/2}}		 

Error

ArcSin[u]+ ArcSin[v] == ArcSin[u*(1 - (v)^(2))^(1/2)+ v*(1 - (u)^(2))^(1/2)]
Missing Macro Error Failure -
Failed [34 / 100]
Result: Complex[4.440892098500626*^-16, 2.633915793849633]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.5707963267948966, -0.6078894033135972]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, 1.5]}

... skip entries to safe data
4.24.E13 Arcsin ⁡ u - Arcsin ⁡ v = Arcsin ⁡ ( u ⁢ ( 1 - v 2 ) 1 / 2 - v ⁢ ( 1 - u 2 ) 1 / 2 ) multivalued-inverse-sine 𝑢 multivalued-inverse-sine 𝑣 multivalued-inverse-sine 𝑢 superscript 1 superscript 𝑣 2 1 2 𝑣 superscript 1 superscript 𝑢 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}u-\operatorname{Arcsin}v=% \operatorname{Arcsin}\left(u(1-v^{2})^{1/2}-v(1-u^{2})^{1/2}\right)}}
\Asin@@{u}-\Asin@@{v} = \Asin@{u(1-v^{2})^{1/2}- v(1-u^{2})^{1/2}}		 

Error

ArcSin[u]- ArcSin[v] == ArcSin[u*(1 - (v)^(2))^(1/2)- v*(1 - (u)^(2))^(1/2)]
Missing Macro Error Failure -
Failed [34 / 100]
Result: Complex[4.440892098500626*^-16, 2.633915793849633]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}


Result: Complex[1.5707963267948966, -0.6078894033135972]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]}

... skip entries to safe data
4.24.E14 Arccos ⁡ u + Arccos ⁡ v = Arccos ⁡ ( u ⁢ v - ( ( 1 - u 2 ) ⁢ ( 1 - v 2 ) ) 1 / 2 ) multivalued-inverse-cosine 𝑢 multivalued-inverse-cosine 𝑣 multivalued-inverse-cosine 𝑢 𝑣 superscript 1 superscript 𝑢 2 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arccos}u+\operatorname{Arccos}v=% \operatorname{Arccos}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)}}
\Acos@@{u}+\Acos@@{v} = \Acos@{uv-((1-u^{2})(1-v^{2}))^{1/2}}		 

Error

ArcCos[u]+ ArcCos[v] == ArcCos[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]
Missing Macro Error Failure -
Failed [63 / 100]
Result: Complex[1.5707963267948963, -3.2418051971632305]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]}


Result: Complex[1.5707963267948966, -3.9508736907744497]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -2]}

... skip entries to safe data
4.24.E14 Arccos ⁡ u - Arccos ⁡ v = Arccos ⁡ ( u ⁢ v + ( ( 1 - u 2 ) ⁢ ( 1 - v 2 ) ) 1 / 2 ) multivalued-inverse-cosine 𝑢 multivalued-inverse-cosine 𝑣 multivalued-inverse-cosine 𝑢 𝑣 superscript 1 superscript 𝑢 2 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arccos}u-\operatorname{Arccos}v=% \operatorname{Arccos}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)}}
\Acos@@{u}-\Acos@@{v} = \Acos@{uv+((1-u^{2})(1-v^{2}))^{1/2}}		 

Error

ArcCos[u]- ArcCos[v] == ArcCos[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]
Missing Macro Error Failure -
Failed [63 / 100]
Result: Complex[-2.3202651922123767, 0.3459279941338042]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.8213274613774166, -2.979843787983438]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.24.E15 Arctan ⁡ u + Arctan ⁡ v = Arctan ⁡ ( u + v 1 - u ⁢ v ) multivalued-inverse-tangent 𝑢 multivalued-inverse-tangent 𝑣 multivalued-inverse-tangent 𝑢 𝑣 1 𝑢 𝑣 {\displaystyle{\displaystyle\operatorname{Arctan}u+\operatorname{Arctan}v=% \operatorname{Arctan}\left(\frac{u+v}{1-uv}\right)}}
\Atan@@{u}+\Atan@@{v} = \Atan@{\frac{u+ v}{1- uv}}		 

Error

ArcTan[u]+ ArcTan[v] == ArcTan[Divide[u + v,1 - u*v]]
Missing Macro Error Failure - Successful [Tested: 1]
4.24.E15 Arctan ⁡ u - Arctan ⁡ v = Arctan ⁡ ( u - v 1 + u ⁢ v ) multivalued-inverse-tangent 𝑢 multivalued-inverse-tangent 𝑣 multivalued-inverse-tangent 𝑢 𝑣 1 𝑢 𝑣 {\displaystyle{\displaystyle\operatorname{Arctan}u-\operatorname{Arctan}v=% \operatorname{Arctan}\left(\frac{u-v}{1+uv}\right)}}
\Atan@@{u}-\Atan@@{v} = \Atan@{\frac{u- v}{1+ uv}}		 

Error

ArcTan[u]- ArcTan[v] == ArcTan[Divide[u - v,1 + u*v]]
Missing Macro Error Failure - Successful [Tested: 1]
4.24.E16 Arcsin ⁡ u + Arccos ⁡ v = Arcsin ⁡ ( u ⁢ v + ( ( 1 - u 2 ) ⁢ ( 1 - v 2 ) ) 1 / 2 ) multivalued-inverse-sine 𝑢 multivalued-inverse-cosine 𝑣 multivalued-inverse-sine 𝑢 𝑣 superscript 1 superscript 𝑢 2 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}u+\operatorname{Arccos}v=% \operatorname{Arcsin}\left(uv+((1-u^{2})(1-v^{2}))^{1/2}\right)}}
\Asin@@{u}+\Acos@@{v} = \Asin@{uv+((1-u^{2})(1-v^{2}))^{1/2}}		 

Error

ArcSin[u]+ ArcCos[v] == ArcSin[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]
Missing Macro Error Failure -
Failed [63 / 100]
Result: Complex[2.3202651922123767, -0.3459279941338042]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[0.8213274613774166, 2.979843787983438]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.24.E16 Arcsin ⁡ u - Arccos ⁡ v = Arcsin ⁡ ( u ⁢ v - ( ( 1 - u 2 ) ⁢ ( 1 - v 2 ) ) 1 / 2 ) multivalued-inverse-sine 𝑢 multivalued-inverse-cosine 𝑣 multivalued-inverse-sine 𝑢 𝑣 superscript 1 superscript 𝑢 2 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}u-\operatorname{Arccos}v=% \operatorname{Arcsin}\left(uv-((1-u^{2})(1-v^{2}))^{1/2}\right)}}
\Asin@@{u}-\Acos@@{v} = \Asin@{uv-((1-u^{2})(1-v^{2}))^{1/2}}		 

Error

ArcSin[u]- ArcCos[v] == ArcSin[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)]
Missing Macro Error Failure -
Failed [63 / 100]
Result: Complex[-1.5707963267948963, 3.2418051971632305]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -1.5]}


Result: Complex[-1.5707963267948966, 3.9508736907744497]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, -2]}

... skip entries to safe data
4.24.E16 Arcsin ⁡ ( u ⁢ v + ( ( 1 - u 2 ) ⁢ ( 1 - v 2 ) ) 1 / 2 ) = Arccos ⁡ ( v ⁢ ( 1 - u 2 ) 1 / 2 - u ⁢ ( 1 - v 2 ) 1 / 2 ) multivalued-inverse-sine 𝑢 𝑣 superscript 1 superscript 𝑢 2 1 superscript 𝑣 2 1 2 multivalued-inverse-cosine 𝑣 superscript 1 superscript 𝑢 2 1 2 𝑢 superscript 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}\left(uv+((1-u^{2})(1-v^{2}))% ^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/2}-u(1-v^{2})^{1/2}% \right)}}
\Asin@{uv+((1-u^{2})(1-v^{2}))^{1/2}} = \Acos@{v(1-u^{2})^{1/2}- u(1-v^{2})^{1/2}}		 

Error

ArcSin[u*v +((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)] == ArcCos[v*(1 - (u)^(2))^(1/2)- u*(1 - (v)^(2))^(1/2)]
Missing Macro Error Failure -
Failed [72 / 100]
Result: Complex[-2.3202651922123763, 0.3459279941338042]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.8213274613774164, -2.979843787983438]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.24.E16 Arcsin ⁡ ( u ⁢ v - ( ( 1 - u 2 ) ⁢ ( 1 - v 2 ) ) 1 / 2 ) = Arccos ⁡ ( v ⁢ ( 1 - u 2 ) 1 / 2 + u ⁢ ( 1 - v 2 ) 1 / 2 ) multivalued-inverse-sine 𝑢 𝑣 superscript 1 superscript 𝑢 2 1 superscript 𝑣 2 1 2 multivalued-inverse-cosine 𝑣 superscript 1 superscript 𝑢 2 1 2 𝑢 superscript 1 superscript 𝑣 2 1 2 {\displaystyle{\displaystyle\operatorname{Arcsin}\left(uv-((1-u^{2})(1-v^{2}))% ^{1/2}\right)=\operatorname{Arccos}\left(v(1-u^{2})^{1/2}+u(1-v^{2})^{1/2}% \right)}}
\Asin@{uv-((1-u^{2})(1-v^{2}))^{1/2}} = \Acos@{v(1-u^{2})^{1/2}+ u(1-v^{2})^{1/2}}		 

Error

ArcSin[u*v -((1 - (u)^(2))*(1 - (v)^(2)))^(1/2)] == ArcCos[v*(1 - (u)^(2))^(1/2)+ u*(1 - (v)^(2))^(1/2)]
Missing Macro Error Failure -
Failed [91 / 100]
Result: Complex[-2.3202651922123767, 2.979843787983438]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}


Result: Complex[-0.8213274613774161, -0.3459279941338048]
Test Values: {Rule[u, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[v, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}

... skip entries to safe data
4.24.E17 Arctan ⁡ u + Arccot ⁡ v = Arctan ⁡ ( u ⁢ v + 1 v - u ) multivalued-inverse-tangent 𝑢 multivalued-inverse-cotangent 𝑣 multivalued-inverse-tangent 𝑢 𝑣 1 𝑣 𝑢 {\displaystyle{\displaystyle\operatorname{Arctan}u+\operatorname{Arccot}v=% \operatorname{Arctan}\left(\frac{uv+1}{v-u}\right)}}
\Atan@@{u}+\Acot@@{v} = \Atan@{\frac{uv+ 1}{v- u}}		 

Error

ArcTan[u]+ ArcCot[v] == ArcTan[Divide[u*v + 1,v - u]]
Missing Macro Error Failure -
Failed [1 / 10]
Result: Indeterminate
Test Values: {Rule[u, Rational[1, 2]], Rule[v, 0.5]}

4.24.E17 Arctan ⁡ u - Arccot ⁡ v = Arctan ⁡ ( u ⁢ v - 1 v + u ) multivalued-inverse-tangent 𝑢 multivalued-inverse-cotangent 𝑣 multivalued-inverse-tangent 𝑢 𝑣 1 𝑣 𝑢 {\displaystyle{\displaystyle\operatorname{Arctan}u-\operatorname{Arccot}v=% \operatorname{Arctan}\left(\frac{uv-1}{v+u}\right)}}
\Atan@@{u}-\Acot@@{v} = \Atan@{\frac{uv- 1}{v+ u}}		 

Error

ArcTan[u]- ArcCot[v] == ArcTan[Divide[u*v - 1,v + u]]
 Missing Macro Error Failure -
Failed [1 / 10]
Result: Indeterminate
Test Values: {Rule[u, Rational[1, 2]], Rule[v, -0.5]}

4.24.E17 Arctan ⁡ ( u ⁢ v + 1 v - u ) = Arccot ⁡ ( v - u u ⁢ v + 1 ) multivalued-inverse-tangent 𝑢 𝑣 1 𝑣 𝑢 multivalued-inverse-cotangent 𝑣 𝑢 𝑢 𝑣 1 {\displaystyle{\displaystyle\operatorname{Arctan}\left(\frac{uv+1}{v-u}\right)% =\operatorname{Arccot}\left(\frac{v-u}{uv+1}\right)}}
\Atan@{\frac{uv+ 1}{v- u}} = \Acot@{\frac{v- u}{uv+ 1}}		 

Error
 
ArcTan[Divide[u*v + 1,v - u]] == ArcCot[Divide[v - u,u*v + 1]]
 Missing Macro Error Failure -
Failed [1 / 10]
Result: Indeterminate
Test Values: {Rule[u, Rational[1, 2]], Rule[v, 0.5]}

4.24.E17 Arctan ⁡ ( u ⁢ v - 1 v + u ) = Arccot ⁡ ( v + u u ⁢ v - 1 ) multivalued-inverse-tangent 𝑢 𝑣 1 𝑣 𝑢 multivalued-inverse-cotangent 𝑣 𝑢 𝑢 𝑣 1 {\displaystyle{\displaystyle\operatorname{Arctan}\left(\frac{uv-1}{v+u}\right)% =\operatorname{Arccot}\left(\frac{v+u}{uv-1}\right)}}
\Atan@{\frac{uv- 1}{v+ u}} = \Acot@{\frac{v+ u}{uv- 1}}		 

Error
 
ArcTan[Divide[u*v - 1,v + u]] == ArcCot[Divide[v + u,u*v - 1]]
 Missing Macro Error Failure -
Failed [1 / 10]
Result: Indeterminate
Test Values: {Rule[u, Rational[1, 2]], Rule[v, -0.5]}

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