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  4. If cos 2 θ+2 sin 2 θ+3 cos 2 θ+4 sin 2 θ+⋅s (200 terms) =10025, where θ is acute, then the value of sin θ- cos θ is:

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Q. If $\cos ^2 \theta+2 \sin ^2 \theta+3 \cos ^2 \theta+4 \sin ^2 \theta+\cdots$ (200 terms) $=10025$, where $\theta$ is acute, then the value of $\sin \theta-\cos \theta$ is:

Trigonometry

Solution:

Given,
$\cos ^2 \theta+2 \sin ^2 \theta+3 \cos ^2 \theta+4 \sin ^2 \theta+\cdots+200 \text { terms }= $
$10025 $
$\Rightarrow(\cos ^2 \theta+3 \cos ^2 \theta+5 \cos ^2 \theta+\cdots 100$ $\text { terms })+(\sin ^2 \theta. $
$.+2 \sin ^2 \theta+\cdots+100 \text { terms })=10025 $
$\Rightarrow 100^2 \cos ^2 \theta+(100^2+100) \sin ^2 \theta=10025 $
$\Rightarrow 10000(\cos ^2 \theta+\sin ^2 \theta)+100 \sin ^2 $
$\theta=10025$
$\Rightarrow 100 \sin ^2 \theta=25 $
$\Rightarrow \sin ^2 \theta=\frac{1}{4} $
$\Rightarrow \sin \theta=\frac{1}{2}(\because \theta \text { is acute })$
$\Rightarrow \theta=30^{\circ} . $
$\therefore \sin \theta-\cos \theta=\sin 30^{\circ}-\cos 30^{\circ} $
$=\frac{1}{2}-\frac{\sqrt{3}}{2}=\frac{1-\sqrt{3}}{2} .$