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  4. cos 2 5°- cos 2 15°- sin 2 15°+ sin 2 35° + cos 15° sin 15°- cos 5° sin 35°=

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Q. $\cos ^{2} 5^{\circ}-\cos ^{2} 15^{\circ}-\sin ^{2} 15^{\circ}+\sin ^{2} 35^{\circ}$ $+\cos 15^{\circ} \sin 15^{\circ}-\cos 5^{\circ} \sin 35^{\circ}=$

AP EAMCETAP EAMCET 2019

Solution:

$\cos ^{2} 5^{\circ}-\cos ^{2} 15^{\circ}-\sin ^{2} 15^{\circ}+\sin ^{2} 35^{\circ}$
$+\cos 15^{\circ} \sin 15^{\circ}-\cos 5^{\circ} \sin 35^{\circ}$
$=\cos 5^{\circ}\left(\cos 5^{\circ}-\sin 35^{\circ}\right)-\cos 15^{\circ}\left(\cos 15^{\circ}-\sin 15^{\circ}\right)$
$+\sin ^{2} 35^{\circ}-\sin ^{2} 15^{\circ}$
$=\cos 5^{\circ}\left(\cos 5^{\circ}-\cos 55^{\circ}\right)-\cos 15^{\circ}\left(\cos 15^{\circ}-\cos 75^{\circ}\right)$
$+\sin 50^{\circ} \sin 20^{\circ}$
$=\cos 5^{\circ}\left(2 \sin 30^{\circ} \sin 25^{\circ}\right)-\cos 15^{\circ}\left(2 \sin 45^{\circ} \sin 30^{\circ}\right)$
$+\sin 50^{\circ} \sin 20^{\circ}$
$=-\frac{\cos 15^{\circ}}{\sqrt{2}}+\cos 5^{\circ} \sin 25^{\circ}+\sin 50^{\circ} \sin 20^{\circ}$
$=-\frac{\cos 15^{\circ}}{\sqrt{2}}+\frac{1}{2}\left(2 \cos 5^{\circ} \cos 65^{\circ}\right)+\frac{1}{2}\left(2 \sin 50^{\circ} \sin 20^{\circ}\right)$
$=-\frac{\cos 15^{\circ}}{\sqrt{2}}+\frac{1}{2}\left[\cos 70^{\circ}+\cos 60^{\circ}+\cos 30^{\circ}-\cos 70^{\circ}\right]$
$=-\frac{\cos 5^{\circ}}{\sqrt{2}}+\frac{1}{2}\left[2 \cos 45^{\circ} \cos 15^{\circ}\right]$
$=-\frac{\cos 15^{\circ}}{\sqrt{2}}+\frac{\cos 15^{\circ}}{\sqrt{2}}=0$