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Find the value of: ( {2{{ }^{ - 1}}{1}{4}} ) + ( {{{ }^{ -

Question

Find the value of: $sin (2 tan^{- 1} \frac{1}{4}) + cos (tan^{- 1} 22)$

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Find the value of:

sin (2 tan^{- 1} \frac{1}{4}) + cos (tan^{- 1} 22)

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Text solutionVerified

Given

sin (2 tan^{- 1} \frac{1}{4}) + cos (tan^{- 1} 22) - - - - (1)

sin (2 tan^{- 1} \frac{1}{4}) = sin ⎝ ⎛ sin^{- 1} \frac{2 ( \frac{1}{4} )}{( \frac{1}{4} )} ⎠ ⎞ [∵ 2 tan^{- 1} x = sin^{- 1} \frac{2 x}{1 + x ^{2}}]

sin (sin^{- 1} \frac{2/4}{17/16})

= sin (sin^{- 1} \frac{8}{17})

= \frac{8}{17}

cos (tan^{- 1} 22)

Let

tan^{- 1} 22 = λ

tan λ = 22

cos λ = \frac{1}{1 + tan ^{2} λ} = \frac{1}{1 + ( 2 2 ) ^{2}} = \frac{1}{1 + 8} = \frac{1}{9}

cos λ = \frac{1}{3}

i.e;

cos (tan^{- 1} 22) = \frac{1}{3}

From

(1) sin (2 tan^{- 1} \frac{1}{4}) + cos (tan^{- 1} 22)

= \frac{8}{17} + \frac{1}{3} = \frac{24 + 17}{51} = \frac{41}{51}

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Question Text	Find the value of: $sin (2 tan^{- 1} \frac{1}{4}) + cos (tan^{- 1} 22)$
Answer Type	Text solution:1
Upvotes	150