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Solve the following equation (^{-1}x)= (^{-1} 34 )

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Solve the following equation $cos (tan^{- 1} x) = sin (cot^{- 1} \frac{3}{4})$

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Solve the following equation

cos (tan^{- 1} x) = sin (cot^{- 1} \frac{3}{4})

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Text solutionVerified

We have,

cos (tan^{- 1} x) = sin (cot^{- 1} \frac{3}{4})

\Rightarrow cos (cos^{- 1} \frac{1}{x ^{2} + 1}) = sin (sin^{- 1} \frac{4}{5})

Let

tan^{- 1} x = θ_{1} \Rightarrow tan θ_{1} = \frac{x}{1}

\Rightarrow cos θ_{1} = \frac{1}{x ^{2} + 1} \Rightarrow θ_{1} = cos^{- 1} \frac{1}{x ^{2} + 1}

And

cot^{- 1} \frac{3}{4} = θ_{2} \Rightarrow cot θ_{2} = \frac{3}{4}

\Rightarrow sin θ_{2} \frac{4}{5} \Rightarrow θ_{2} sin^{- 1} \frac{4}{5}

\Rightarrow \frac{1}{x ^{2} + 1} = \frac{4}{5}

{∵ cos (cos^{- 1} x) = x, x \in [- 1, 1] an d sin (sin^{- 1} x) = x, x \in [- 1, 1]}

On squaring both sides, we get

\Rightarrow 16 (x^{2} + 1) = 25

\Rightarrow 15 x^{2} = 9

\Rightarrow x^{2} = (\frac{3}{4})^{2}

∴ x = \pm \frac{3}{4} = \frac{- 3}{4}, \frac{3}{4}

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Question Text	Solve the following equation $cos (tan^{- 1} x) = sin (cot^{- 1} \frac{3}{4})$
Answer Type	Text solution:1
Upvotes	150