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
B
p2q2
C
p2q2p2+q2
D
p2+q2p2q2

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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