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({}{4}+{1}{2} ^{-1}x)+ ({}{4}-{1}{2} ^{-1}x) equals

Question

$tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} x) + tan (\frac{π}{4} - \frac{1}{2} cos^{- 1} x)$ equals

A

\frac{1}{x}

B

x

C

\frac{2}{x}

D

2 x

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tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} x) + tan (\frac{π}{4} - \frac{1}{2} cos^{- 1} x)

equals

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Text solutionVerified

We have,

tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} x) + tan (\frac{π}{4} - \frac{1}{2} cos^{- 1} x)

l e t cos^{- 1} x = t \Rightarrow x = cos t

\Rightarrow

tan (\frac{x}{4} + \frac{t}{2}) + tan (\frac{x}{4} - \frac{t}{2})

= \frac{tan ( \frac{π}{4} ) + tan ( \frac{t}{2} )}{1 - tan ( \frac{π}{4} ) tan ( \frac{t}{2} )} + \frac{tan ( \frac{π}{4} ) - tan ( \frac{t}{2} )}{1 + tan ( \frac{π}{4} ) tan ( \frac{t}{2} )}

= \frac{1 + tan ( \frac{t}{2} )}{1 - tan ( \frac{t}{2} )} + \frac{1 - tan ( \frac{t}{2} )}{1 + tan ( \frac{t}{2} )}

= \frac{[ 1 + tan ( \frac{t}{2} ) ] ^{2} + [ 1 - tan ( \frac{t}{2} ) ] ^{2}}{( 1 - tan ( \frac{t}{2} )) ( 1 + tan ( \frac{t}{2} ) )}

= \frac{2 ( 1 + tan ^{2} ( \frac{t}{2} ))}{1 - tan ^{2} ( \frac{t}{2} )}

= \frac{2}{cos t}

= \frac{2}{x}

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Question Text	$tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} x) + tan (\frac{π}{4} - \frac{1}{2} cos^{- 1} x)$ equals
Answer Type	Text solution:1
Upvotes	150