SUPPLEMENTARY PROBLEMS

1. Prove the following

1. = 2 cosec A   
2. sec⁶A - tan ⁶A = 1 + 3tan²A + 3 tan⁴A

3. (1+ sin + cos ) ² = 2 (1 + sin ) (1 + cos )    
4. sin ⁶ + cos⁶ = 1 - 3cos² + 3 cos⁴

5. sin x ( cot x + 3 ) ( 3 cot x + 1 ) = 3 cosec x + 10 cos x

6. If x = a cos² + b sin² prove that (x- a ) (b - x ) = ( a - b )² sin² cos²

7. If sin = Find cos and cot

8. ( sin - cos ) ( sin + cos ) = - cos2

9. (tan²A - cot A)²= tan ⁴A - 2 tan 2A cot A + cot²A

10. sec - sec sin² = cos

11. tan A sin A + cot A cos A = ( sec A + cosec A) (sec A. cosec A - 1) . sin A cos A

12. 

13. 

1. 

2. 

3. 

4. 

5. tan - csc sec ( 1 - cos²) = cot

6. (x sin - y cos )² + (x cos + y sin ) = x²+ y²

7. 

3. Find the values of t- ratios of , given that tan = 5/4

Ans. Quad. I:-

Quad III:-

4.

1. Prove that

2. Prove that

3. Prove that

4. Prove that cos 225⁰ x cos 675⁰ + sin 585⁰
   sin 315⁰ = 0

5. Evaluate

6. Prove that sin 20⁰ sin 40⁰sin 60⁰sin 80 ⁰ = 3/6 ( Don't use calculator or table)

7. Show that sin a sin (60 ⁰ - a) sin ( 60⁰ + a) = 1/4 sin 3a

8. Prove that

9. Prove that

10. 1) If sin 3a = 0.4 Find cos 3a

    2) Find sin a if tan a /2 = 1/

11. Prove that (cos A - cos B) ² + (sin A - sin B)² = 4 sin ²

12. If A + B + C = 180 ⁰ Prove that cos A+ cos B - cos C = 4 cos A/2 cos B/2 sin C/2 -1

13. When A + B + C = 180⁰, show that sin 2A + sin 2B + sin 2C = 4 sin A . sin B sin C

5. Find the values of sin (a + b), cos (a + b) and tan (a + b), given

1. sin a = 8/17, tan b= 5/12 a , and b in Quad I.

   Ans. 171/221, 140/221, -63/16

2. sin a = 1/3, sin b = 2 /5, a in Quad I b in Quad II,

   Ans.

6. Find the values of sin ( a - b ) , cos (a - b) and tan (a - b ) given

1. cos a = -12/13 , cot b = 24/7, a in Quad. II, b in Quad I

   Ans. 204/325, -253/325, 21/220

2. sin a = 1/3, sin = 2/5, a in Quad. II, b in Quad I

Ans.

7. Prove that tan (45⁰+ q) =

2. sin (a + b) sin (a - b) ) = sin²a - sin²b) and sin (a + b) ) sin (a -b) ) = cos²b - cos²a

3. sin 50⁰ - cos 80 ⁰ = cos 700
4. sin 40⁰ - cos 70⁰ = cos 80⁰

5. cos 20⁰ cos 40⁰ cos 60⁰ cos 80⁰ = 1 / 16
6. 

7. sin 75⁰ - cos 15⁰ = cos 105⁰ + sin 15⁰

8. cos²x + cos²(x +120⁰) + cos²(x - 120⁰) = 3/2.

8. Find the values of sin 2 , cos 2 and tan 2 ; given

1. sin = 3/5, is in Quad. I

   Ans. 24/25, 7/25, 24/7

2. tan = - 1/5, is in Quad. II

   Ans. - 6/13, 12/13, - 5/12

3. Prove that 2 using x=30⁰

4. Prove that using A = 60⁰

9. Find the maximum and minimum value of each sum and value of x or t between 0 x or t 180⁰

1. 4 cos x + 3 sin x into the form (sin (x-a))
   Ans. max. 5 when x = 36 ⁰ 22^'
   min -5 when x = 216 ⁰ 52^'

2. 5 cos 3t + 12 sin 3t into the form cos (3t - a)
   Ans . max 13 when t = 22⁰ 38^'
   min -13 when t = 82⁰38^'

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Index

Trignometric Identities

4.1 Fundamental Identities
4.2 The addition formulas
4.3 The multiple-angle (Double & Half angle) formulas
4.4 Tangent Identities
4.5 Factorization & Defactorization

Supplementary Problems