Chapter 21 - Trigonometrical Identities (Including Trigonometrical Ratios of Complementary Angles and Use of Four Figure Trigonometrical Tables) Exercise Ex. 21(E)

Question 1

Prove the following identities:

(i) 

(ii) 

(iii) 

(iv) 

(v) 

(vi) 

(vii) 

(viii) 

(ix) 

(x) 

(xi) 

(xii) 

(xiii) 

(xiv) 

(xv) 

(xvi) 

(xvii) 

Solution 1

(i) 

 

 

 

 

(ii)

 

 

 

(iii) 

 

 

 

(iv) 

 

 

 

(v) 

 

 

 

(vi) 

 

 

(vii) 

 

 

 

(viii) 

 

 

 

 

 

 

 

 

 

 

 

(ix) 

 

(x) 

 

 

(xi)

 

 

 

(xii) 

 

 

 

(xiii) 

 

 

 

(xiv) 

 

 

 

(xv) 

 

 

 

(xvi) 

 

 

 

(xvii) 

 

 

Question 2

If and , then prove that:

q(p² - 1) = 2p

Solution 2

 

 

Question 3

If , show that:

 

Solution 3

 

 

 

Question 4

If , show that:

 

 

Solution 4

 

 

Question 5

If tan A = n tan B and sin A = m sin B, prove that:

 

 

Solution 5

 

 

 

 

Question 6

(i) If 2 sinA - 1 = 0, show that:

sin 3A = 3 sinA - 4 sin³A

(ii) If 4 cos²A - 3 = 0, show that:

cos 3A = 4 cos³A - 3 cosA

 

Solution 6

(i) 2 sinA - 1 = 0

 

 

(ii)

 

 

 

 

 

 

Question 7

Evaluate:

(i) 

(ii) 

(iii) 

(iv) 

(v) 

(vi) 

(vii) 

(viii) 

 

 

 

 

 

 

 

 

Solution 7

(i) 

 

 

 

 

(ii) 

 

 

 

 

 

(iii) 

 

 

 

 

(iv) 

 

 

 

 

(v) 

 

 

 

(vi) 

 

 

 

(vii) 

 

 

 

(viii) 

 

 

Question 8

Prove that:

(i) 

(ii) 

(iii) 

(iv) 

(v) 

 

 

 

Solution 8

(i) 

(ii) 

(iii) 

 

 

 

(iv) 

 

 

 

(v) 

 

 

Question 9

If A and B are complementary angles, prove that:

(i) 

(ii) 

(iii) cosec²A + cosec²B = cosec²A cosec²B

(iv) 

 

 

Solution 9

Since, A and B are complementary angles, A + B = 90°

(i)

 

 

 

(ii)

 

 

 

 

 

(iii)

= cosec²A [sec(90 - B)]²

= cosec²A cosec²B

 

 

 

 

 

 

 

 

(iv) 

 

 

 

 

 

 

 

 

 

 

 

 

Question 10

Prove that:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution 10

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Question 11

If 4cos²A - 3 = 0 and 0°A 90°, then prove that:

(i) sin3A= 3 sinA - 4 sin³A

(ii) cos3A= 4 cos³A - 3 cosA

Solution 11

4 cos²A - 3 = 0

 

 

 

 

 

 

 

 

 

Question 12

Find A, if 0°A 90° and:

(i) 

(ii) sin 3A - 1 = 0

(iii) 

(iv) 

(v) 

Solution 12

(i) 

 

 

(ii) sin 3A - 1 = 0

 

 

 

(iii) 

 

 

 

(iv) 

 

 

(v) 

 

 

 

Question 13

If 0° < A < 90°; find A, if:

(i) 

 

 

(ii) 

 

Solution 13

(i) 

 

 

 

(ii) 

 

 

 

 

 

Question 14

Prove that:

(cosec A - sin A) (sec A - cos A) sec²A = tan A

Solution 14

Question 15

Prove the identity (sin θ + cos θ) (tan θ + cot θ) = sec θ + cosec θ.

Solution 15

Question 16

Evaluate without using trigonometric tables,

sin² 28° + sin² 62° + tan² 38° - cot² 52° + sec² 30° 

Solution 16

sin² 28° + sin² 62° + tan² 38° - cot² 52° + sec² 30° 

= sin² 28° + [sin (90 - 28)°]² + tan² 38° - [cot(90 - 38)°]² + sec² 30° 

= sin² 28°  + cos² 28° + tan² 38° - tan² 38° + sec² 30°