Prove that sec2 (tan-1 2) + cosec2 (cot-1 3) = 15

1.	Prove that sec2 (tan-1 2) + cosec2 (cot-1 3) = 15
Answer» Let tan^-12 = θ ⇒ tan θ = 2 sec² θ = 1 + tan² θ = 1 + (2)² = 1 + 4 = 5 ∴ sec² (tan^-12) = 5 Let cot^-13 = Φ = cot Φ = 3 ∴ cosec² Φ = 1 + cot² Φ = 1 + (3)² = 1 + 9 = 10 ∴ cosec²(cot^-1 3) = 10 …..(ii) Adding eq. (i) and (ii), we get sec² (tan^-12) + cosec² (cot^-13) = 5 + 10 ⇒ sec² (tan^-12) + cosec²(cot^-13) = 15 Hence proved.

Answer»

Let tan^-12 = θ

⇒ tan θ = 2

sec² θ = 1 + tan² θ

= 1 + (2)²

= 1 + 4 = 5

∴ sec² (tan^-12) = 5

Let cot^-13 = Φ = cot Φ = 3

∴ cosec² Φ = 1 + cot² Φ

= 1 + (3)² = 1 + 9 = 10

∴ cosec²(cot^-1 3) = 10 …..(ii)

Adding eq. (i) and (ii), we get

sec² (tan^-12) + cosec² (cot^-13) = 5 + 10

⇒ sec² (tan^-12) + cosec²(cot^-13) = 15

Hence proved.

Discussion