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If ^{ -1 }{ { { 1+{ x }^{ 2 } } - { 1-{ x }^{ 2 } } }{ {

Question

If $tan^{- 1} \frac{1 + x ^{2} - 1 - x ^{2}}{1 + x ^{2} + 1 - x ^{2}} = α$ , then $x^{2} =$

A

cos 2 α

B

sin 2 α

C

tan 2 α

D

cot 2 α

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Sebastian Discussed

If

tan^{- 1} \frac{1 + x ^{2} - 1 - x ^{2}}{1 + x ^{2} + 1 - x ^{2}} = α

, then

x^{2} =

12 mins ago

Discuss this question LIVE

12 mins ago

Text solutionVerified

Since

tan^{- 1} \frac{1 + x ^{2} - 1 - x ^{2}}{1 + x ^{2} + 1 - x ^{2}} = α

\Rightarrow \frac{1 + x ^{2} - 1 - x ^{2}}{1 + x ^{2} + 1 - x ^{2}} = \frac{tan α}{1}

Using componendo and dividendo, we get

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

\Rightarrow \frac{1 + x ^{2}}{1 - x ^{2}} = \frac{tan ^{2} ( \frac{π}{4} + α )}{1}

Again using componendo and dividendo, we get

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

\Rightarrow x^{2} = - cos (\frac{π}{2} + 2 α) = sin 2 α

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Question Text	If $tan^{- 1} \frac{1 + x ^{2} - 1 - x ^{2}}{1 + x ^{2} + 1 - x ^{2}} = α$ , then $x^{2} =$
Answer Type	Text solution:1
Upvotes	150