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Question

If $$\cot\theta + \cos\theta = p$$ and $$\cot\theta - \cos\theta = q$$, then the value of $$p^2 - q^2$$ is


A
2pq
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B
4pq
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C
2pq
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D
4pq
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Solution

The correct option is B $$4\sqrt{pq}$$
$$p^{2}-q^{2}$$
$$=(cot\theta+cos\theta)^{2}-(cot\theta-cos\theta)^{2}$$
$$=(cos^{2}\theta+cot^{2}\theta+2cos\theta cot\theta)-(cos^{2}+cot^{2}\theta-2cos\theta cot\theta)$$
$$=4cos\theta.cot\theta$$
$$=\dfrac{4cos^{2}\theta}{sin\theta}$$
$$=4\sqrt{\dfrac{cos^{4}\theta}{sin^{2}\theta}}$$

$$=4\sqrt{\dfrac{{cos^{2}\theta}\times{cos^{2}\theta}}{sin^{2}\theta}}$$

$$=4\sqrt{\dfrac{{cos^{2}\theta}\times(1-{sin^{2}\theta})}{sin^{2}\theta}}$$

$$=4\sqrt{cot^{2}\theta-cos^{2}\theta}$$
$$=4\sqrt{(cot\theta+cos\theta)(cot\theta-cos\theta)}$$
$$=4\sqrt{pq}$$

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