If (sinθ-cosθ)/(sinθ+cosθ) = 4/5 , then the value of (cosec²θ)/(2-cosec²θ) is :

If (sinθ-cosθ)/(sinθ+cosθ) = 4/5 , then the value of…

If $\frac{sinθ-cosθ}{sinθ+cosθ}$ = $\frac{4}{5}$ , then the value of $\frac{cosec^2θ}{2-cosec^2θ}$ is :

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Correct answer: C

Explanation

The correct answer is option 3 i.e. $\frac{41}{40}$ .

(sinθ - cosθ)/(sinθ + cosθ) = 4/5

⇒ 5sinθ - 4sinθ = 4cosθ + 5cosθ

⇒ sinθ = 9cosθ

⇒ sinθ/cosθ = 9 = tanθ

⇒ sec²θ = 1 + tan²θ = 1 + 9² = 1 + 81 = 82

⇒ cos²θ = 1/82

⇒ sin²θ = 1 - cos²θ = 1 - 1/82 = 81/82

⇒ cosec²θ = 82/81

Now, cosec²θ/(2 - cosec²θ) = (82/81)/(2 - 82/81) = (82/81)/(80/81) = 41 / 40

Therefore, the required value is 41/40.