Prove the following trigonometric identities. If 3 +5 =5, prove t…

Question

Prove the following trigonometric identities.
If $3\sin\theta+5\cos\theta=5,$ prove that $5\sin\theta-3\cos\theta=\pm3.$

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Answer

Given, $3\sin\theta+5\cos\theta=5$
We have to prove that $5\sin \theta-3\cos\theta=5$
We know that $\sin^2\theta+\cos^2\theta=1$
$\text{L.H.S}=3\sin\theta+5\cos\theta=5$
Squaring the given equation, we have
$(3\sin\theta+5\cos\theta)^2=(5)^2$
$\Rightarrow\ 9\sin^2\theta+2\times3\sin\theta\times5\cos\theta+25\cos^2\theta=25$
$\Rightarrow\ 9(1-\cos^2\theta)+2\times3\sin\theta\times5\cos\theta+25(1-\sin^2\theta)=25$
$\Rightarrow\ 9-9\cos^2\theta+2\times3\sin\theta\times5\cos\theta+25-25\sin^2\theta=25$
$\Rightarrow\ 34-(9\cos^2\theta-2\times3\sin\theta\times5\cos\theta+25\sin^2\theta)=25$
$\Rightarrow\ -(25\sin^2\theta-2\times5\sin\theta\times3\cos\theta+9\cos^2\theta)=-9$
$\Rightarrow\ (25\sin^2\theta-2\times5\sin\theta\times3\cos\theta+9\cos^2\theta)=9$
$\Rightarrow\ (5\sin\theta-3\cos\theta)^2=9$
$\Rightarrow\ 5\sin\theta-3\cos\theta=\pm3=\text{R.H.S}$
$\therefore\ \text{L.H.S}=\text{R.H.S}$

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