# P Given 4 Cos θ − Sin θ 2 Cos θ + Sin θ What is the Value of C O S E C 2 θ − Sec 2 θ C O S E C 2 θ + Sec 2 θ - Mathematics

Sum

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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#### APPEARS IN

RD Sharma Class 10 Maths
Chapter 10 Trigonometric Ratios
Q 8 | Page 55