Chapter 9: Trigonometric Ratios of Multiple and Sub Multiple Angles – Exercise 9.2

{∵ sin 3θ = 3 sin θ – 4 sin³ θ cos 3θ = 4 cos³ θ – 3 cos θ}

= 1/4 [3(sin 3θ cos θ + sin θ cos 3θ) + cos 3θ sin 3θ – sin 3θ cos 3θ]

= 1/4 [3 sin (3θ + θ) + 0]

= 3/4 sin 4θ

So,

cos³ θ sin³θ + sin³θ cos 3θ = 3/4 sin 4θ

 

Trigonometric Ratios of Multiple and Sub Multiple Angles – Exercise 9.2 – Q.4

We have to prove that

sin 5A = 5cos⁴ A sin A - 10cos² A sin³A + sin⁵ A

L.H.S = sin 5A = sin (3A + 2A)

= sin 3A cos 2A + cos 3A. sin 2A

= (3sin A - 4sin³ A) (2cos² A -1) + (4cos³ A - 3cos A) 2sin A cos A.

= -3 sin A + 4 sin³ A + 6 sin A cos²A - 8 sin³A cos²A + 8 cos⁴A sin A - 6 cos²A sin A

= 8cos⁴A sin A – 8 sin³A cos²A – 3sin A + 4 sin³A

= 5 cos⁴A sin A - 10 sin³A cos²A – 3 sin A + 3 cos⁴A sin A + 4 sin³A + 2 sin³A cos²A

= 5 cos⁴A sin A – 10 sin³A cos²A – 3 sin A (1 -cos⁴A) + 2 sin³A (2 + cos²A)

= 5cos⁴A sin A - 10 sin³A cos²A – 3 sin A (1- cos² A) (1 + cos²A) +2 sin³ A(2 + cos²A)

= 5 cos⁴A sin A- 10 sin³A cos²A – 3 sin³ A (1 + cos²A) + 2 sin³ A (2 + cos²A)

= 5cos⁴AsinA-10 sin³Acos²A - sin³ A [3(1 + cos²A) - 2(2 + cos²A)]

= 5 cos⁴A sin A- 10 sin³A cos²A - sin³A [3 + 3 cos²A - 4 – 2cos²A]

= 5 cos⁴A sin A - 10 sin³A cos²A - sin³A [cos²A - 1]

= 5 cos⁴A sin A - 10 sin³A cos²A + sin⁵A

= RHS

 

Trigonometric Ratios of Multiple and Sub Multiple Angles – Exercise 9.2 – Q.5

tan A × tan(A + 60°) + tan A × tan(A - 60°) + tan(A + 60°) tan(A - 60°)

 

Trigonometric Ratios of Multiple and Sub Multiple Angles – Exercise 9.2 – Q.6

tan A + tan(60° + A) - tan(60° - A) = 3 tan 3A

LHS = tan A + tan (60° + A) - tan (60° - A)

= 3 tan 3A

So,

tan A + tan(60° + A) - tan(60° - A) = 3 tan 3A

 

Trigonometric Ratios of Multiple and Sub Multiple Angles – Exercise 9.2 – Q.7

LHS = cot A + cot(60° + A) - cot(60° - A)

= 3cot 3A

= RHS

LHS = RHS

Hence proved

 

Trigonometric Ratios of Multiple and Sub Multiple Angles – Exercise 9.2 – Q.8

LHS = cotA + cot(60° + A) + cot(120° + A)

= cotA + cot(60° + A) - cot[180° - (120° + A)]

{since - cotθ = cot(180° - θ)}

= cotA + cot(60° + A) - cot(60° - A)

= 3cot 3A

LHS = RHS

 

Trigonometric Ratios of Multiple and Sub Multiple Angles – Exercise 9.2 – Q.9

= RHS

LHS = RHS