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

Given:

\( 3 \sin \theta+5 \cos \theta=5 \)

To do:

We have to prove that \( 5 \sin \theta-3 \cos \theta=\pm 3 \).

Solution:

We know that,

$\sin^2 A+\cos^2 A=1$

Therefore,

$3 \sin \theta+5 \cos \theta=5$

Squaring on both sides, we get,

$(3 \sin \theta+5 \cos \theta)^2=(5)^2$

$\Rightarrow 9 \sin ^{2} \theta+25 \cos ^{2} \theta+30 \sin \theta \cos \theta=25$ $\Rightarrow 9\left(1-\cos ^{2} \theta\right)+25\left(1-\sin ^{2} \theta\right)+30 \sin \theta \cos \theta=25$

$\Rightarrow 9-9 \cos ^{2} \theta+25-25 \sin ^{2} \theta+30 \sin \theta \cos \theta=25$

$\Rightarrow -25 \sin ^{2} \theta-9 \cos ^{2} \theta+30 \sin \theta \cos \theta=25-9-25$

$\Rightarrow -25 \sin ^{2} \theta-9 \cos ^{2} \theta+30 \sin \theta \cos \theta=-9$

$\Rightarrow  25 \sin ^{2} \theta+9 \cos ^{2} \theta-30 \sin \theta \cos \theta=9$

$\Rightarrow (5 \sin \theta)^{2}+(3 \cos \theta)^{2}-2 \times 5 \sin \theta \times 3 \cos \theta=(\pm 3)^{2}$

$\Rightarrow (5 \sin \theta-3 \cos \theta)^{2}=(\pm 3)^{2}$

$\Rightarrow 5 \sin \theta-3 \cos \theta=\pm 3$

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Updated on: 10-Oct-2022

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