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Question
If \[\tan \theta = \frac{1}{\sqrt{7}}, \text{ then } \frac{{cosec}^2 \theta - \sec^2 \theta}{{cosec}^2 \theta + \sec^2 \theta} =\]
Options
\[\frac{5}{7}\]
\[\frac{3}{7}\]
\[\frac{1}{12}\]
\[\frac{3}{4}\]
Solution
Given that:
`tan θ=1/sqrt7`
We are asked to find the value of the following expression
`(cosec^2θ-sec^2θ)/(cosec^2θ+sec^2θ)`
Since `tan θ= "Perpendicular"/"Base"` .
⇒ `"Perpendicular"=1`
⇒ `"Base"= sqrt7`
⇒ `"Hypotenuse"=sqrt(1+7)`
⇒`" Hypotenuse"=sqrt8`
We know that `secθ="Hypotenuse"/"Base" and cosecθ= "Hypotenuse"/"Perpendicular"`
We find:
`(Cosec^2θ-sec^2 θ)/(Cosec^2 +sec^2 θ)`
`((sqrt8/1)^2-(sqrt8/sqrt7)^2)/((sqrt8/1)^2+(sqrt8/sqrt7)^2)`
=(8/1-8/7)/(8/1+8/7)
=`(48/7)/(64/7)`
=`3/4`
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