Question:
If 3 sin x + 5 cos x = 5, then write the value of 5 sin x − 3 cos x.
Solution:
$3 \sin x+5 \cos x=5$ (Given)
Squaring both the sides:
$9 \sin ^{2} x+25 \cos ^{2} x+30 \sin x \cos x=25$
$30 \sin x \cos x=25-9 \sin ^{2} x-25 \cos ^{2} x$ (1)
We have to find the value of $5 \sin \theta-3 \cos \theta$.
$(5 \sin x-3 \cos \mathrm{x})^{2}=25 \sin ^{2} \mathrm{x}+9 \cos ^{2} x-30 \sin x \cos x$
$(5 \sin x-3 \cos x)^{2}=25 \sin ^{2} x+9 \cos ^{2} x-\left(25-9 \sin ^{2} x-25 \cos ^{2} x\right) \quad[$ From (1) $]$
$(5 \sin x-3 \cos x)^{2}=34 \sin ^{2} x+34 \cos ^{2} x-25$
$(5 \sin x-3 \cos x)^{2}=34-25 \quad\left(\because \sin ^{2} x+\cos ^{2} x=1\right)$
$(5 \sin x-3 \cos x)^{2}=9$
$5 \sin x-3 \cos x=\pm 3$