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Inverse Trigonometric Functions

Question

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$cos to the power of negative 1 end exponent x plus sin to the power of negative 1 end exponent fraction numerator 1 over denominator square root of 5 end fraction equals straight pi over 4$

Solution

cos to the power of negative 1 end exponent straight x plus sin to the power of negative 1 end exponent fraction numerator 1 over denominator square root of 5 end fraction equals straight pi over 4

rightwards double arrow space space cos to the power of negative 1 end exponent straight x equals straight pi over 4 minus sin to the power of negative 1 end exponent fraction numerator 1 over denominator square root of 5 end fraction space space space space space space space space space rightwards double arrow space cos to the power of negative 1 end exponent straight x equals cos to the power of negative 1 end exponent space fraction numerator 1 over denominator square root of 2 end fraction minus cos to the power of negative 1 end exponent space square root of 1 minus 1 fifth end root rightwards double arrow space space cos to the power of negative 1 end exponent space straight x equals cos to the power of negative 1 end exponent space fraction numerator 1 over denominator square root of 2 end fraction minus cos to the power of negative 1 end exponent space fraction numerator 2 over denominator square root of 5 end fraction rightwards double arrow space space cos to the power of negative 1 end exponent straight x equals cos to the power of negative 1 end exponent space open square brackets fraction numerator 1 over denominator square root of 2 end fraction cross times fraction numerator 1 over denominator square root of 5 end fraction plus square root of open parentheses 1 minus 1 half close parentheses open parentheses 1 minus 4 over 5 close parentheses end root close square brackets space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because 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 space cos to the power of negative 1 end exponent open parentheses straight x space straight y plus square root of 1 minus straight x squared end root square root of 1 minus straight y squared end root close parentheses close square brackets rightwards double arrow space space cos to the power of negative 1 end exponent space straight x space equals space cos to the power of negative 1 end exponent space open parentheses fraction numerator 2 over denominator square root of 10 end fraction plus fraction numerator 1 over denominator square root of 10 end fraction close parentheses space space space rightwards double arrow space space straight x equals fraction numerator 2 over denominator square root of 10 end fraction plus fraction numerator 1 over denominator square root of 10 end fraction equals fraction numerator 3 over denominator square root of 10 end fraction

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