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1. Prove that sec^{2} (tan^{-1} 2) + cosec^{2} (cot^{-1} 3) = 15

Sec^{2} (tan^{-1} 2) + cosec^{2} (cot^{-1} 3)

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

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

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Plus 2 Biology Science

ICSE/ISC