Inverse Trigonometric Functions

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Question

Prove that $cos to the power of negative 1 end exponent 4 over 5 plus cos to the power of negative 1 end exponent 12 over 13 equals cos to the power of negative 1 end exponent 12 over 13 equals cos to the power of negative 1 end exponent 33 over 65.$

Solution

cos open square brackets cos to the power of negative 1 end exponent 4 over 5 plus cos to the power of negative 1 end exponent 12 over 13 close square brackets equals cos open parentheses cos to the power of negative 1 end exponent 4 over 5 close parentheses cos open parentheses cos to the power of negative 1 end exponent 12 over 13 close parentheses minus sin open parentheses cos to the power of negative 1 end exponent 4 over 5 close parentheses sin open parentheses cos to the power of negative 1 end exponent 12 over 13 close parentheses equals c o s open parentheses c o s to the power of negative 1 end exponent 4 over 5 close parentheses c o s open parentheses c o s to the power of negative 1 end exponent 12 over 13 close parentheses minus square root of 1 minus cos squared open parentheses cos to the power of negative 1 end exponent 4 over 5 close parentheses end root square root of 1 minus cos squared open parentheses cos to the power of negative 1 end exponent 12 over 13 close parentheses end root equals 4 over 5 cross times 12 over 13 minus square root of 1 minus open parentheses 4 over 5 close parentheses squared end root square root of 1 minus open parentheses 13 over 13 close parentheses squared end root equals 4 over 5 cross times 12 over 13 minus 3 over 5 cross times 5 over 13 equals 48 over 65 minus 15 over 65 equals 33 over 65 therefore space space cos space open square brackets cos to the power of negative 1 end exponent 4 over 5 plus cos to the power of negative 1 end exponent 12 over 13 close square brackets equals 33 over 65 space space space space rightwards double arrow space space cos to the power of negative 1 end exponent 4 over 5 plus cos to the power of negative 1 end exponent 12 over 13 equals cos to the power of negative 1 end exponent open parentheses 33 over 65 close parentheses

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Inverse Trigonometric Functions

Prove that $cos to the power of negative 1 end exponent 4 over 5 plus cos to the power of negative 1 end exponent 12 over 13 equals cos to the power of negative 1 end exponent 12 over 13 equals cos to the power of negative 1 end exponent 33 over 65.$

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