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Question
If θ is an acute angle such that \[\tan^2 \theta = \frac{8}{7}\] then the value of \[\frac{\left( 1 + \sin \theta \right) \left( 1 - \sin \theta \right)}{\left( 1 + \cos \theta \right) \left( 1 - \cos \theta \right)}\]
Options
\[\frac{7}{8}\]
\[\frac{8}{7}\]
\[\frac{7}{4}\]
\[\frac{64}{49}\]
Solution
Given that: `tan^2 θ=8/7` and θis an acute angle
We have to find the following expression `((1+sinθ)(1-sin θ))/((1+cos θ)(1-cos θ))`
Since
`tan^2θ=8/7`
`tan θ=sqrt(8/7)`
`tan θ=sqrt8/sqrt7`
Since `tan θ="perpendiular"/"Base"`
⇒ `"Perpendicular"=sqrt8`
⇒ `"Base"=sqrt7`
⇒ `"Hypotenuse"
= sqrt(8+7)`
⇒ `"Hypotenuse"=sqrt15`
We know that `sinθ= "Perpendicular"/"Hypotenuse" and cos θ="Base"/"Hypotenuse"`
We find:
`((1+sinθ )(1-sin θ))/((1+cos θ)(1-cosθ))`
=`((1+sqrt8/sqrt15)(1-sqrt8/sqrt15))/((1+sqrt7/sqrt15)(1-sqrt7/sqrt15))`
=`((1-8/15))/((1-7/15))`
=`(7/15)/(8/15)`
=`7/8`
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