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

(b)
(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.

Solution

About

KS Teng

KS has been teaching H2/H1 Mathematics and IB mathematics for almost two decades. Having taught students from all various Junior Colleges, KS adapts to students' abilities and help them better understand the topics. As someone who loves to teach mathematics and sees it as a truly useful tool in life, KS seeks to enable students to appreciate math. Therefore, his tuition mission is to motivate and cultivate students to be independent and confident thinkers.

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Interesting Trigonometry Question

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