Question

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

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Answer 1

Answer:

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Step-by-step explanation:

Answer 2

Answer:

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)

= a² = 1 + 2(sin x)(cos x)

= (sin x)(cos x) = (a² - 1)/2

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 - ((a² - 1)/2)²)

Now, (sin² x + cos² x)² = sin⁴ x + cos⁴x + 2(sin²x)(cos2 x)

= sin⁴ x + cos⁴ x = 1 - 2((a² - 1)/2)²

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

sin⁶ x + cos⁶ x = 1 - 2((a² - 1)/2)² - ((a² - 1)/2)²

= 1 - 3(a² - 1)²/4

= (4 - 3a⁴ - 3 + 6a²)/4

= (1 + 6a² - 3a⁴)/4

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