Sequences and Series 5 Question 15
6. If $\cos (x-y), \cos x$ and $\cos (x+y)^{\prime}$ are in HP. Then $\cos x \cdot \sec \frac{y}{2}=\ldots$.
(1997C, 2M)
Analytical & Descriptive Questions
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Answer:
Correct Answer: 6. $\pm \sqrt{2}$
Solution:
- Since, $\cos (x-y), \cos x$ and $\cos (x+y)$ are in HP.
$$ \begin{aligned} & \therefore \quad \cos x=\frac{2 \cos (x-y) \cos (x+y)}{\cos (x-y)+\cos (x+y)} \\ & \Rightarrow \quad \cos x(2 \cos x \cdot \cos y)=2{\cos ^{2} x-\sin ^{2} y } \\ & \Rightarrow \quad \cos ^{2} x \cdot \cos y=\cos ^{2} x-\sin ^{2} y \\ & \Rightarrow \quad \cos ^{2} x(1-\cos y)=\sin ^{2} y \\ & \Rightarrow \quad \cos ^{2} x \cdot 2 \sin ^{2} \frac{y}{2}=4 \sin ^{2} \frac{y}{2} \cdot \cos ^{2} \frac{y}{2} \\ & \Rightarrow \quad \cos ^{2} x \cdot \sec ^{2} \frac{y}{2}=2 \\ & \therefore \quad \cos x \cdot \sec \frac{y}{2}= \pm \sqrt{2} \end{aligned} $$
