Question

If sinx +cosx=c then sin^6x+cos^6x=?

Answer 1

hope it helps,any doubt then comment

Answer 2

Answer: (1 + 6c2 - 3c4)/4

Step-by-step explanation:

(sin x + cos x)2 = sin2 x + cos2 x + 2(sin x)(cos x)

⇨ (sin x + cos x)2 = 1 + 2(sin x)(cos x)

⇨ c2 = 1 + 2(sin x)(cos x)

⇨ (sin x)(cos x) = (c2 - 1)/2

Now, Using the identity

a3 + b3 = (a + b)(a2 − ab + b2), we have  

(sin2 x)3 + (cos2 x)3 = (sin2 x + cos2 x)(sin4 x + cos4 x - (sin2 x)(cos2 x))

= (sin4 x + cos4 x - ((c2 - 1)/2)2)

Now, (sin2 x + cos2 x)2 = sin4 x + cos4 x + 2(sin2 x)(cos2 x)

⇨ sin4 x + cos4 x = 1 - 2((c2 - 1)/2)2  

Using the value of sin4 x + cos4 x, we have  

sin6 x + cos6 x = 1 - 2((c2 - 1)/2)2 - ((c2 - 1)/2)2  

= 1 - 3(c2 - 1)2/4

= (4 - 3c4 - 3 + 6c2)/4

= (1 + 6c2 - 3c4)/4