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Inverse Trigonometric Functions

Question

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Show that $sin to the power of negative 1 end exponent 3 over 5 minus sin to the power of negative 1 end exponent 8 over 17 equals cos to the power of negative 1 end exponent 84 over 85.$

Solution

Let space sin to the power of negative 1 end exponent 3 over 5 equals straight x space and space sin to the power of negative 1 end exponent 8 over 17 equals straight y therefore space space space space space space sin space straight x equals 3 over 5 space and space sin space straight y space equals space 8 over 17 Now space space space space cos space straight x equals square root of 1 minus sin squared space straight x end root equals square root of 1 minus 9 over 25 end root equals square root of 16 over 25 end root equals 4 over 5 and space space space space space space cos space straight y equals square root of 1 minus sin squared space straight y end root equals square root of 1 minus 64 over 289 end root equals square root of 225 over 289 end root equals 15 over 17

Now cos(x - y) = cos x cos y + sin x sin y

equals 4 over 5 cross times 15 over 17 plus 3 over 5 cross times 8 over 17 equals 60 over 85 plus 24 over 85 equals fraction numerator 60 plus 24 over denominator 85 end fraction equals 84 over 85 therefore space space space space space straight x minus straight y equals cos to the power of negative 1 end exponent 84 over 85 sin to the power of negative 1 end exponent 3 over 5 asterisk times sin to the power of negative 1 end exponent 8 over 17 minus cos to the power of negative 1 end exponent 84 over 85

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