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Question
If θ is an acute angle such that sec2 θ = 3, then the value of \[\frac{\tan^2 \theta - {cosec}^2 \theta}{\tan^2 \theta + {cosec}^2 \theta}\]
Options
\[\frac{4}{7}\]
\[\frac{3}{7}\]
\[\frac{2}{7}\]
\[\frac{1}{7}\]
Solution
Given that:
`sec^2θ=3`
`secθ=sqrt3`
We need to find the value of the expression
`(tan^2θ-cosec^2θ)/(tan^2θ+cosec^2θ)`
`"since" secθ="Hypotenuse"/"Base"`. so
⇒` "Hypotenuse"= sqrt3`
⇒ `"Base"=1`
⇒ `"Perpendicular"=sqrt(3-1)`
⇒ `"Perpendicular"=sqrt2`
Here we have to find: `(tan^2θ-cosec^2θ)/(tan^2θ+cosec^2θ)`
⇒`(tan^2θ-cosec^2θ)/(tan^2θ+cosec^2θ) = (2/1-3/2)/(2/1+3/2)`
⇒`(tan^2θ-cosec^2θ)/(tan^2θ+cosec^2θ)=(1/2)/(7/2)`
⇒`(tan^2θ-cosec^2θ)/(tan^2θ+cosec^2θ)=1/7`
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