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Inverse Trigonometric Functions

Question

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Prove that $cos to the power of negative 1 end exponent 12 over 13 plus sin to the power of negative 1 end exponent 3 over 5 equals sin to the power of negative 1 end exponent 56 over 65.$

Solution

sin open square brackets cos to the power of negative 1 end exponent 12 over 13 plus sin to the power of negative 1 end exponent 3 over 5 close square brackets equals space sin open parentheses cos to the power of negative 1 end exponent 12 over 13 close parentheses cos open parentheses sin to the power of negative 1 end exponent 3 over 5 close parentheses plus cos open parentheses c o s to the power of negative 1 end exponent 12 over 13 close parentheses c o s open parentheses s i n to the power of negative 1 end exponent 3 over 5 close parentheses equals 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 square root of 1 minus sin squared open parentheses sin to the power of negative 1 end exponent 3 over 5 close parentheses end root plus cos open parentheses cos to the power of negative 1 end exponent 13 over 13 close parentheses sin open parentheses sin to the power of negative 1 end exponent 3 over 5 close parentheses equals space square root of 1 minus open parentheses 12 over 13 close parentheses squared end root square root of 1 minus open parentheses 3 over 5 close parentheses squared end root plus 12 over 13 cross times 3 over 5 equals 5 over 13 cross times 4 over 5 plus 12 over 13 cross times 3 over 5 equals 20 over 65 plus 36 over 65 equals 56 over 65 therefore space space space sin open square brackets cos to the power of negative 1 end exponent 12 over 13 plus sin to the power of negative 1 end exponent 3 over 5 close square brackets equals 56 over 65 space space rightwards double arrow space space cos to the power of negative 1 end exponent 12 over 13 plus sin to the power of negative 1 end exponent 3 over 5 equals sin to the power of negative 1 end exponent 3 over 5 equals sin to the power of negative 1 end exponent open parentheses 56 over 65 close parentheses

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