SATHEE: Sequences and Series 5 Question 15

Sequences and Series 5 Question 15

6. If $\cos (x - y), \cos x$ and $\cos (x + y)^{'}$ are in HP. Then $\cos x \cdot \sec \frac{y}{2} = \dots$ .

(1997C, 2M)

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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.

$\begin{aligned} ∴ \cos x = \frac{2 \cos (x - y) \cos (x + y)}{\cos (x - y) + \cos (x + y)} \\ \Rightarrow \cos x (2 \cos x \cdot \cos y) = 2 \cos^{2} x - \sin^{2} y \\ \Rightarrow \cos^{2} x \cdot \cos y = \cos^{2} x - \sin^{2} y \\ \Rightarrow \cos^{2} x (1 - \cos y) = \sin^{2} y \\ \Rightarrow \cos^{2} x \cdot 2 \sin^{2} \frac{y}{2} = 4 \sin^{2} \frac{y}{2} \cdot \cos^{2} \frac{y}{2} \\ \Rightarrow \cos^{2} x \cdot \sec^{2} \frac{y}{2} = 2 \\ ∴ \cos x \cdot \sec \frac{y}{2} = \pm \sqrt{2} \end{aligned}$

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Sequences and Series 5 Question 15

6. If cos⁡(x−y),cos⁡x and cos⁡(x+y)′ are in HP. Then cos⁡x⋅sec⁡y2=….

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6. If $\cos (x - y), \cos x$ and $\cos (x + y)^{'}$ are in HP. Then $\cos x \cdot \sec \frac{y}{2} = \dots$ .