Trigonometrical Equations 1 Question 28
28. Find the values of $x(-\pi, \pi)$ which satisfy the equation $2^{1+|\cos x|+\left|\cos ^{2} x\right|+\ldots}=4$
(1984, 2M)
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Answer:
Correct Answer: 28. ${x: x=n \pi} \cup \quad x: x=n \pi+(-1)^{n} \frac{\pi}{10}$ U $x: x=n \pi+(-1)^{n} \quad \frac{-3 \pi}{10}$
Solution:
- Given, $2^{1+|\cos x|+\left|\cos ^{2} x\right|+\left|\cos ^{3} x\right|+\ldots .}=2^{2}$
$$ \begin{array}{ll} \Rightarrow & 2^{\frac{1}{1-|\cos x|}}=2^{2} \\ \Rightarrow & \frac{1}{1-|\cos x|}=2 \end{array} $$
$$ \begin{array}{ll} \Rightarrow & |\cos x|=\frac{1}{2} \\ \Rightarrow & \cos x= \pm \frac{1}{2} \\ \therefore & x=\frac{\pi}{3}, \frac{2 \pi}{3},-\frac{\pi}{3},-\frac{2 \pi}{3} \quad[\because x \in(-\pi, \pi)] \end{array} $$
Thus, the solution set is $\pm \frac{\pi}{3}, \pm \frac{2 \pi}{3}$.
