Show that sum of 9 Sin4 θ - 6 Sin6 θ and 9 Cos4 θ - 6 Cos6 θ  remains constant for all values of θ.

Asked by nareshpanchal76 | 7th Oct, 2020, 10:13: AM

Expert Answer:

To find the sum of 9sin4θ - 6sin6θ and 9cos4θ - 6cos6θ 
9sin4θ - 6sin6θ + 9cos4θ - 6cos6θ = 9(sin4θ + cos4θ) - 6(cos6θ + sin6θ)
= 9[(sin2θ + cos2θ)2 - 2sin2θ cos2θ] - 6(sin2θ + cos2θ)(sin4θ + cos4θ - sin2θcos2θ)
= 9[1 - 2sin2θ cos2θ] - 6[(sin2θ + cos2θ)2 - 2sin2θ cos2θ - sin2θcos2θ]
= 9[1 - 2sin2θ cos2θ] - 6[1 - 3sin2θ cos2θ]
= 9 - 18sin2θ cos2θ - 6 + 18sin2θ cos2θ
= 9 - 6
= 3

Answered by Renu Varma | 30th Oct, 2020, 10:57: AM