First slide

Multiple and sub- multiple Angles

Question

If $\frac{\tan 3 A}{\tan A} = k (k \neq 1)$ then which of the following is not true?

Moderate

A

$\frac{\cos A}{\cos 3 A} = \frac{k - 1}{2}$
B

$\frac{\sin 3 A}{\sin A} = \frac{2 k}{k - 1}$
C

$\frac{\cot 3 A}{\cot A} = \frac{1}{k}$
D

none of these

Solution

$\begin{array}{l} \frac{\tan 3 A}{\tan A} = k (k \neq 1) \\ ∴ \frac{\sin 3 Acos A}{\sin Acos 3 A} = \frac{k}{1} \\ ∴ \frac{k + 1}{k - 1} = \frac{\sin 4 A}{\sin 2 A} = 2 \cos 2 A (using comp. and divd.) \\ ∴ \frac{\sin 3 A}{\sin A} = 1 + 2 \cos 2 A = \frac{2 k}{k - 1} \end{array}$
Similarly, $\frac{\cos A}{\cos 3 A} = \frac{k - 1}{2}$

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