First slide

Multiple and sub- multiple Angles

Question

If $\tan A \cdot \tan B = \frac{1}{2}, then (5 - 3 \cos 2 A) (5 - 3 \cos 2 B) =$

Easy

A

2
B

8
C

12
D

16

Solution

$\begin{array}{l} \tan A \cdot \tan B = \frac{1}{2} \\ \Rightarrow \frac{\tan A}{1} = \frac{\cot B}{2} = λ (say) \\ \Rightarrow \tan A = λ and \tan B = \frac{1}{2 λ} \\ ∴ (5 - 3 \cos 2 A) (5 - 3 \cos 2 B) \\ = [5 - \frac{3 (1 - λ^{2})}{(1 + λ^{2})}] \times [5 - \frac{3 \cdot (1 - \frac{1}{4 λ^{2}})}{(1 + \frac{1}{4 λ^{2}})}] = 16 \end{array}$

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