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

Sec²  (tan^-1 2) + cosec² (cot^-1 3)

= sec² (sec^-1 √5) + cosec² (cosec^-1 √10)

= {sec (sec^-1 √5 )}² + {cosec² (cosec^-1 √10) }²

= (√5)² + (√10)²

= 5 + 10 = 15

2. Prove that : sin (2 tan^{-1 3}/5 - sin^{-1 7}/25) = ³⁰⁴/425

3. Solve for x : sin (2 tan^-1 x) = 1

Sin (2 tan^-1 x) = 1

4. Solve : cos^-1 [sin (cos^-1 x)] = ^π/3

cos^-1 [sin (cos^-1 x)] = ^π/3

5. Solve for x : tan^-1 (x - 1) + tan^-1 x + tan^-1 (x + 1) = tan^-1 3x

tan^-1 (x - 1) + tan^-1 x + tan^-1 (x + 1) = tan^-1 3x

6. Prove that : 2 tan^-1 (¹/3) + cot^-1 (4) = tan^-1 (¹⁶/13)

7. Show  that : sin^-1 (¹/√17) + cos^-1 (⁹/√85) = tan^-1 (¹/2)

Let θ = sin^{-1 1}/√17 

∴  sin θ = ¹/√17

∴ tan θ = ¹/4 or θ = tan^{-1 1}/4  and  α = cos^{-1 9}/√85

∴ cos α = ⁹/√85

∴ tan α = ²/9  or  α = tan^{-1 2}/9

8. Prove that 2 (tan^-1 1 + tan^{-1 1}/2 + tan^{-1 1}/3) = π

9. Show that sin^{-1 4}/5 + cos^{-1 2}/√5 = cot^{-1 2}/11

sin^{-1 4}/5 = tan^{-1 4}/3 and cos^{-1 2}/√5 = tan^{-1 1}/2

10. Prove : 2 sin^{-1 3}/5 = tan^{-1 24}/7

Let sin^{-1 3}/5 = x

Then, sin x = ³/5