Interesting Trigonometry Question

We know how A’levels like to combine a few topics across. We also know how bad trigonometry can be. I was coaching my International Bachelorette (IB) class last week and came across a fairly interesting question. It tests students on their abilities to manage double angle identities. Not a tough question, but definitely good practice 🙂
FYI: All double angle identities are found in MF26.
Here it is…

(a) Show that (1 + i \text{tan} \theta)^n + (1 - i \text{tan} \theta)^n = \frac{2 \text{cos} n \theta}{ \text{cos}^n \theta}, \text{cos} \theta \neq 0.

(i) Use the double angle identity \text{tan} 2 \theta = \frac{2 \text{tan} \theta}{1 - \text{tan}^2 \theta} to show that \text{tan} \frac{\pi}{8} = \sqrt{2} - 1.

(ii) Show that \text{cos} 4x = 8 \text{cos}^4 x - 8 \text{cos}^2 x + 1.

(iii) Hence find the value of \int^{\frac{\pi}{8}}_0 {\frac{2 \text{cos} 4x}{ \text{cos}^2 x}}~dx exactly.

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