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Question
Given
\[\frac{4 \cos \theta - \sin \theta}{2 \cos \theta + \sin \theta}\] what is the value of \[\frac{{cosec}^2 \theta - \sec^2 \theta}{{cosec}^2 \theta + \sec^2 \theta}\]
Solution
Given: ` tan θ= 1/sqrt5`
We know that: `tan θ=("Prependicular")/("Base")`
`("Prependicular")/("Base")=1/sqrt5`
`"Hypotenuse"= sqrt( ("Perpendicular")^2+("Base")^2)`
`"Hypotenuse"=sqrt(1+5)`
`"Hypotenuse"=sqrt6`
Now we find, `(cosec^2θ-sec^2θ)/(cosec^2θ+sec^2θ)`
=`(("hypotenuse")^2/("Perpendicular")^2-("hypotenuse")^2/("Base")^2)/(("hypotenuse")^2/("Perpendicular")^2+("hypotenuse")^2/("Base")^2)`
= `((sqrt6)^2/(1)^2-(sqrt6)^2/(sqrt5)^2)/((sqrt6)^2/(1)^2+((sqrt6))/(sqrt5)^2)`
= `(6/1-6/5)/(6/1+6/5)`
=`(24/5)/(36/5)`
=`2/3`
Hence the value of `(cosec^2θ-sec^2θ)/(cosec^2θ+sec^2θ)` is `2/3`
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