If sin x + cos x = c, then sin^6 x + cos^6 x is equal to

If sin x + cos x = c, then sin6 x + cos6 x is equal to

  1. (1 + 6c2 - 3c4)/16
  2. (1 + 6c2 - 3c4)/4
  3. (1 + 6c2 + 3c4)/16
  4. (1 + 6c2 + 3c4)/4

Answer

Consider (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

The correct option is B.