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{ cos }^{ -1 }( { 3+5 { x } }{ 5+3 { x } } ) =

Question

$cos^{- 1} (\frac{3 + 5 cos x}{5 + 3 cos x}) =$

A

t an^{- 1} (\frac{1}{2} tan \frac{x}{2})

B

2 tan^{- 1} (- \frac{1}{2} tan \frac{x}{2})

C

\frac{1}{2} tan^{- 1} (2 tan \frac{x}{2})

D

2 tan^{- 1} (\frac{1}{2} tan \frac{x}{2})

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Text solutionVerified

Let

cos^{- 1} \frac{( 3 + 5 cos x )}{( 5 + 3 cos x )} = α

\frac{3 + 5 cos x}{5 + 3 cos x} = cos α

∵ cos α \frac{[ 1 - tan ^{2} ( \frac{α}{2} )]}{[ 1 + tan ^{2} ( \frac{x}{2} )]}

cos α = 5 \frac{\frac{[ 1 - tan ^{2} ( \frac{x}{2} )]}{[ 1 + tan ^{2} ( \frac{x}{2} )]} + 3}{5 + 3 [ \frac{1 - tan ^{2} ( \frac{x}{2} )}{1 + tan ^{2} ( \frac{x}{2} )} ]}

\Rightarrow \frac{5 ( 1 - tan ^{2} ( \frac{x}{2} )) + 3 ( 1 + tan ^{2} ( \frac{x}{2} ))}{5 ( 1 + tan ^{2} ( \frac{x}{2} )) + 3 ( 1 - tan ^{2} ( \frac{x}{2} ))}

\Rightarrow \frac{5 - 5 tan ^{2} ( \frac{x}{2} ) + 3 + 3 tan ^{2} ( \frac{x}{2} )}{5 + 5 tan ^{2} ( \frac{x}{2} ) + 3 - 3 tan ^{2} ( \frac{x}{2} )}

\Rightarrow \frac{8 - 2 tan ^{2} ( \frac{x}{2} )}{8 + 2 tan ^{2} ( \frac{x}{2} )}

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

∴ sin α = 1 - cos^{2} α

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

\Rightarrow \frac{[ 4 + tan ^{2} ( \frac{x}{2} ) ^{2} ] [ 4 + tan ^{2} ( \frac{x}{2} ) ]}{4 + tan ^{2} ( \frac{x}{2} )}^{2}^{2}

= \frac{4 + tan ^{2} ( \frac{x}{2} ) - 4 + tan ^{2} ( \frac{x}{2} ) ( 4 + tan ^{2} ( \frac{x}{2} + 4 - tan ^{2} ) ( \frac{x}{2} ) )}{[ 4 + tan ^{2} ( \frac{x}{2} ) ] ^{2}}

\Rightarrow \frac{2 tan ^{2} ( \frac{x}{2} ) .8}{4 + tan ^{2} ( \frac{x}{2} )} \Rightarrow \frac{4 tan ( \frac{x}{2} )}{4 tan ^{2} ( \frac{n}{2} )}

∵ tan (\frac{x}{2}) = \frac{1 - cos α}{sin α}

∴ \frac{1 - ⎝ ⎛ \frac{4 - tan ^{2} ( \frac{n}{2} )}{4 + tan ^{2} ( \frac{n}{2} )} ⎠ ⎞}{⎝ ⎛ \frac{4 tan ( \frac{n}{2} )}{4 + tan ^{2} ( \frac{n}{2} )} ⎠ ⎞} \Rightarrow \frac{4 + tan ^{2} ( \frac{n}{2} - 4 + tan ^{2} ) ( \frac{n}{2} )}{4 tan ( \frac{n}{2} )}

tan (\frac{x}{2}) \Rightarrow \frac{2 tan ^{2} ( \frac{x}{2} )}{2 ^{4} tan ( \frac{x}{2} )} \Rightarrow \frac{1}{2} tan (\frac{x}{2})

\frac{α}{2} = tan^{- 1} [\frac{1}{2} tan (\frac{x}{2})]

cos^{- 1} (\frac{5 cos x + 3}{5 + 3 cos x}) \Rightarrow 2 tan^{- 1} [\frac{1}{2} tan (\frac{n}{2})]

So, option (b) is correct.

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Question Text	$cos^{- 1} (\frac{3 + 5 cos x}{5 + 3 cos x}) =$
Answer Type	Text solution:1
Upvotes	151