#### Topics

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 Paid Users Only!
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