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Question

If $$\sec\theta = \displaystyle\frac{\sqrt{p^2 + q^2}}{q}$$, then the value of $$\displaystyle\frac{p\sin\theta - q\cos\theta}{p\sin\theta + q\cos\theta}$$


A
pq
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B
p2q2
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C
p2q2p2+q2
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D
p2+q2p2q2
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Solution

The correct option is C $$\displaystyle\frac{p^2 - q^2}{p^2+q^2}$$
$$\sec\theta=\cfrac{hypotenuse}{base}=\cfrac{\sqrt{p^{2}+q^{2}}}{q}$$
Hence,
$$hypotenuse=\sqrt{p^{2}+q^{2}}$$
$$base=q$$
Therefore by applying, Pythagoras theorem, we get the altitude as
$$p$$.
Hence
$$\sin\theta=\cfrac{p}{\sqrt{p^{2}+q^{2}}}$$
$$\cos\theta=\cfrac{q}{\sqrt{p^{2}+q^{2}}}$$
Hence substituting, the values in the above question we get
$$\cfrac{p\sin\theta-q\cos\theta}{p\sin\theta+q\cos\theta}$$
$$=\cfrac{p^{2}-q^{2}}{p^{2}+q^{2}}$$

Mathematics

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