Inverse Trigonometric Functions

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Question

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

Solution

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

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