Trigonometrical Ratios and Identities 2 Question 3
3. Suppose $\sin ^{3} x \sin 3 x=\sum _{m=0}^{n} C _m \cos n x$ is an identity in $x$, where $C _0, C _1, \ldots, C _n$ are constants and $C _n \neq 0$. Then, the value of $n$ is… .
(1981, 2M)
Show Answer
Answer:
Correct Answer: 3. 6
Solution:
- Given, $\sin ^{3} x \sin 3 x=\sum _{m=0}^{n} C _m \cos n x$ is an identity in $x$, where, $C _0, C _1, \ldots, C _n$ are constants. $\sin ^{3} x \sin 3 x=\frac{1}{4}{3 \sin x-\sin 3 x} \cdot \sin 3 x$
$$ =\frac{1}{4} \frac{3}{2} \cdot 2 \sin x \cdot \sin 3 x-\sin ^{2} 3 x $$
$$ \begin{aligned} & =\frac{1}{4} \frac{3}{2}(\cos 2 x-\cos x)-\frac{1}{2}(1-\cos 6 x) \\ & =\frac{1}{8}(\cos 6 x+3 \cos 2 x-3 \cos x-1) \end{aligned} $$
$\therefore$ On comparing both sides, we get $n=6$
