Question

If sinx + sin^2x = 1 then, show that cos^12x + 3cos^10x + 3cos^8x + cos^6x = 1

Answer 1

This can be proved by using identities of trigonometry.

Prove that cos¹²x + 3cos¹⁰x + 3cos⁸x + cos⁶x = 1:

Given that,  

sin x + sin²x = 1 ⇒ sin x = 1 - sin²x

⇒ sin x = cos²x   (∵sin²x + cos²x = 1)

We have to prove that,

cos¹²x + 3cos¹⁰x + 3cos⁸x + cos⁶x = 1

LHS = cos¹²x + 3cos¹⁰x + 3cos⁸x + cos⁶x

= sin⁶x + 3sin⁵x + 3sin⁴ + sin³x

= (sin²x)³ + 3(sin²x)²sin x + 3(sin²x) (sin x)² + (sin x)³

= (sin²x + sin x)³           (∵(a+b)³ = a³ + 3a²b + 3 ab² + b³)

= 1³ = 1 = RHS

LHS = RHS

Hence the equation is proved since both sides of the equation are equal.

Learn more about trigonometric identities here:

brainly.com/question/1979770

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