Question

Given that cot θ = 1/√5, what is the value of (sec²θ - cosec²θ)/(sec²θ + cosec²θ) ?

(a) 2/3
(b) 3/2
(c) 25/36
(d) 12/13​

Answer 1

Explanation:

`\mathsf{Given :\;\frac{{sec}^2\theta - co{sec}^2\theta}{{sec}^2\theta + co{sec}^2\theta}}`

`\bigstar\;\;\textsf{We know that : \large\boxed{\mathsf{{sec}\theta = (1)/(cos\theta)}}}`

`\bigstar\;\;\textsf{We know that : \large\boxed{\mathsf{co{sec}\theta = (1)/(sin\theta)}}}`

`\mathsf{\implies ((1)/(cos^2\theta) - (1)/(sin^2\theta))/((1)/(cos^2\theta) + (1)/(sin^2\theta))}`

`\mathsf{\implies ((sin^2\theta - cos^2\theta)/(sin^2\theta.cos^2\theta))/((sin^2\theta + cos^2\theta)/(sin^2\theta.cos^2\theta))}`

`\mathsf{\implies (sin^2\theta - cos^2\theta)/(sin^2\theta + cos^2\theta)}`

Taking sin²θ common in both numerator & denominator, We get :

`\mathsf{\implies (sin^2\theta\left(1 - (cos^2\theta)/(sin^2\theta)\right))/(sin^2\theta\left(1 + (cos^2\theta)/(sin^2\theta)\right))}`

`\bigstar\;\;\textsf{We know that : \large\boxed{\mathsf{cot\theta = (cos\theta)/(sin\theta)}}}`

`\mathsf{\implies (1 -cot^2\theta)/(1 + cot^2\theta)}`

`\mathsf{Given :\;cot\theta = (1)/(√(5))}`

`\mathsf{\implies (1 - \left((1)/(√(5))\right)^2)/(1 + \left((1)/(√(5))\right)^2)}`

`\mathsf{\implies (1 - (1)/(5))/(1 + (1)/(5))}`

`\mathsf{\implies ((5 - 1)/(5))/((5 + 1)/(5))}`

`\mathsf{\implies (5 - 1)/(5 + 1)}`

`\mathsf{\implies (4)/(6)}`

`\mathsf{\implies (2)/(3)}`

Hence, option (a) 2/3 is your correct answer.