Question

If tanθ = (1/√5), find the value of (cosec²θ – sec²θ)/(cosec²θ + sec²θ) .

Answer 1

(cosec² θ - sec² θ) / (cosec² θ + sec² θ) = 2 / 3  

Explanation:

tanθ = (1/√5)

tan² θ= 1/ 5 ---------(1)

As tan² θ = sec² θ - 1, we get:

sec² θ - 1 = 1/5

sec² θ = 1/5 + 1 = 6/5

Taking the reciprocal on both sides of equation (1), we get:

1/ tan² θ= 5

Cot ² θ = 5

As Cot² θ = Cosec² θ - 1, we get:

Cosec² θ - 1 = 5

Cosec² θ = 6

(cosec² θ - sec² θ) / (cosec² θ + sec² θ) = (6 - 6/5) / (6 + 6/5)

= (24/5) / (36/5)

= 24 / 36

= 2 / 3