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Inverse Trigonometric Functions

Question
CBSEENMA12032729
Wired Faculty App

Prove that cos to the power of negative 1 end exponent x plus cos to the power of negative 1 end exponent y equals cos to the power of negative 1 end exponent open square brackets x y minus square root of left parenthesis 1 minus x squared right parenthesis left parenthesis 1 minus y squared right parenthesis end root close square brackets

Solution

Put cos-1x=straight theta, cos-1y=straight ϕ
therefore  x=cos straight theta, y=cos straight ϕ
Now space cos space left parenthesis straight theta plus straight ϕ right parenthesis equals cosθ space cosϕ minus sin space straight theta space sin space straight theta therefore space space cos left parenthesis straight theta plus straight ϕ right parenthesis equals cos space straight theta space cos space straight ϕ minus square root of 1 minus cos squared straight theta end root space square root of 1 minus cos squared space straight ϕ end root therefore space space left parenthesis straight theta plus straight ϕ right parenthesis equals cos to the power of negative 1 end exponent open square brackets cos space straight theta space cos space straight ϕ minus square root of 1 minus cos squared straight theta end root space square root of 1 minus cos squared space straight ϕ end root close square brackets rightwards double arrow space space cos to the power of negative 1 end exponent straight x plus cos to the power of 1 space straight y equals cos to the power of negative 1 end exponent open square brackets straight x space straight y minus square root of 1 minus straight x squared end root square root of 1 minus straight y squared end root close square brackets cos to the power of negative 1 end exponent straight x plus cos to the power of 1 space straight y equals cos to the power of negative 1 end exponent open square brackets straight x space straight y minus square root of left parenthesis 1 minus straight x squared right parenthesis end root square root of left parenthesis 1 minus straight y squared right parenthesis end root close square brackets

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