Question

Prove that sin²x + cos²(x + π/6) + cos²(x - π/6) = 3/2

Answer 1

Step-by-step explanation:

cos²x + cos²(x + π/3) + cos²(x - π/3)

= cos²x + (cosx cosπ/3 - sinx sinπ/3)² + (cosx cosπ/3 + sinx sinπ/3)²

= cos²x + (1/2 cosx - √3 /2 sinx)² + (1/2 cosx + √3/2 sinx)²

= cos²x + 1/4 cos²x + 3/4 sin²x - 2 * 1/2 *√3/2 cosx sinx + 1/4 cos²x + 3/4 sin²x + 2 * 1/2 *√3/2 cosx sinx

= cos²x + 1/2 cos²x + 3/2 sin²x

= 3/2 cos²x + 3/2 sin²x

= 3/2 (cos²x + sin²x)

=3/2

Hence proved

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