Question 16

Evaluate $$\left[\cos^2 \left(\frac{\pi}{32}\right) + \cos^2 \left(\frac{3 \pi}{32}\right) + \cos^2 \left(\frac{5 \pi}{32}\right) + ... + \cos^2 \left(\frac{15 \pi}{32}\right)\right] - \left[\sin^2 \left(\frac{\pi}{16}\right) + \sin^2 \left(\frac{2 \pi}{16}\right) + ... + \sin^2 \left(\frac{7 \pi}{16}\right)\right]$$

Solution

$$\cos\left(A\right)\ =\sin\left(\frac{\pi}{2}-A\right)$$

$$\sin A=\cos\left(\frac{\pi}{2}-A\right)$$

$$\cos\left(\frac{\pi}{32}\right)=\sin\left(\frac{\pi}{2}-\frac{\pi}{32}\right)=\sin\left(\frac{15\pi}{32}\right)$$

$$\sin^2\left(A\right)+\cos^2\left(A\right)=1$$

So the series simplifies to (1+1+1+1)-(1+1+1+1/2)

therefore value of series is 1/2

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