A geometry problem by Md Zuhair

Geometry Level 2

True or False?

\quad tan ⁡ − 1 x = sin ⁡ − 1 x cos ⁡ − 1 x \tan^{-1} x = \dfrac{\sin^{-1} x}{\cos^{-1} x } is an identity.

Restrict to the domain where these functions are all defined.

True False

Md Zuhair by Md Zuhair , 20, India

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1 solution

Chew-Seong Cheong
Oct 24, 2016

When x = 1 x=1 , we have the L H S = tan ⁡ − 1 1 = π 4 LHS = \tan^{-1} 1 = \dfrac \pi 4 , while the R H S = sin ⁡ − 1 1 cos ⁡ − 1 1 = π 2 2 π = 1 4 ≠ L H S RHS = \dfrac {\sin^{-1}1}{\cos^{-1}1} = \dfrac {\frac \pi 2}{2 \pi} =\dfrac 14 \ne LHS . Therefore, the answer is False \boxed{\text{False}} ; the expression is not an identity.

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