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Derivative of Some Standard Functions

Prove : cos ∧...

Question

Prove : cos^2 π/8+cos^2 3π/8+cos^2 5π/8+cos^2 7π/8=2

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Solution

cos²(π/8) + cos²(3π/8) + cos²(5π/8) + cos²(7π/8)
= cos²(π/8) + cos²(π/2 - π/8) + cos²(5π/8) + cos²(π/2 - (-3π/8))
= cos²(π/8) + sin²(π/8) + cos²(5π/8) + sin²(-3π/8)
= cos²(π/8) + sin²(π/8) + cos²(5π/8) + sin²(5π/8) -----> sin(-3π/8) = -sin(5π/8) but after squaring it becomes positive anyway
= 1 + 1
= 2

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\cos^{2} \frac{π}{8} + \cos^{2} \frac{3 π}{8} + \cos^{2} \frac{5 π}{8} + \cos^{2} \frac{7 π}{8} = 2

c o s^{2} \frac{π}{8} + c o s^{2} \frac{3 π}{8} + c o s^{2} \frac{5 π}{8} + c o s^{2} \frac{7 π}{8} = 2

Q. Prove that :

cos 4 A = 1 - 8 {cos}^{2} A + 8 {cos}^{4} A

Q. Prove the following identities:

{cos}^{2} \frac{π}{8} + {cos}^{2} \frac{3 π}{8} + {cos}^{2} \frac{5 π}{8} + {cos}^{2} \frac{7 π}{8} = 2

\cos 4 x = 1 - 8 \cos^{2} x + 8 \cos^{4} x

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