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[tex] \scriptsize\colorbox{lightyellow} {\text{ \bf♕ Brainliest answer }}[/tex][tex] \rule{300pt}{0.1pt}[/tex][tex]\mathbb\red{ \tiny A \scriptsize \: N \small \:S \large \: W \Large \:E \huge \: R}[/tex][tex] \rule{300pt}{0.1pt}[/tex]

Question :-

sin 6x + sin 4x - sin 2x = 4 cos x sin 2x cos 3x

Solution :-

[tex]\begin{array}{l}\rm L.H.S. =\sin 6 x+\sin 4 x-\sin 2 x \\ \\ \displaystyle \rm =2 \sin \left(\frac{6 x+4 x}{2}\right) \cos \left(\frac{6 x-4 x}{2}\right)-2 \sin x \cos x \\ \\ \rm\displaystyle\rm -2 \sin 5 x \cos x-2 \sin x \cos x \\ \\ \rm\displaystyle\rm =2 \cos x(\sin 5 x-\sin x) \\ \\ \rm\displaystyle\rm =2 \cos x\left[2 \cos \left(\frac{5 x+x}{2}\right) \sin \left(\frac{5 x-x}{2}\right)\right] \\ \\ \rm\displaystyle\rm =2 \cos x(2 \cos 3 x \sin 2 x) \\ \\ \rm\displaystyle \rm =4 \cos x \sin 2 x \cos 3 x \\ \\ \rm\displaystyle \rm =R.H.S. \end{array}[/tex]

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Question :-

sin20° sin40° sin60° sin80° = 3/16

Solution :-

[tex] \rm \operatorname{LHS}= \sin 20^{\circ} \cdot \sin 40^{\circ} \cdot \sin 60 \cdot \sin 80^{\circ} [/tex]

[tex]\sin 60^{\circ}\left[\sin 20^{\circ} \cdot \sin (60-20) \cdot \sin (60+\right. 20) [/tex]

[tex]\[ \begin{array}{l} \\ \displaystyle\rm =\frac{\sqrt{3}}{2}\left[\frac{\sin 3(20)}{4}\right] \\ \\ \\ \displaystyle\rm=\frac{\sqrt{3}}{2}\left[\frac{\sin 60^{\circ}}{4}\right] \\\\ \\ \displaystyle\rm =\frac{\sqrt{3}}{2}\left[\frac{\sqrt{3}}{2} \times \frac{1}{4}\right] \\\\ \\ \displaystyle\rm =\frac{\sqrt{3}}{2}\left[\frac{\sqrt{3}}{8}\right] \\ \\ \\ \displaystyle\rm=\frac{3}{16} \qquad\qquad [ \because\sqrt{3} \times \sqrt{3} = 3 ] \\\\ \\ \displaystyle\rm \text { LHS }=\text { RHS } \end{array} \][/tex]

Hence, proved.

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Question :-

cos20° cos40° cos60° cos80° = 1/16

Solution :-

[tex]\[ \begin{array}{l} \\ \\ \displaystyle\rm \text { LHS } = \cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ} \cos 60^{\circ} \\ \\ \\ \displaystyle\rm =\cos 20^{\circ} \cos \left(60^{\circ}-20^{\circ}\right) \cos \left(60^{\circ}+20^{\circ}\right) \cos 60^{\circ} \\ \\ \\ \displaystyle\rm =\frac{1}{4} \cos \left(3 \times 20^{\circ}\right) \cos 60^{\circ} \\ \\ \\ \displaystyle\rm =\frac{1}{4} \cos ^{2} 60^{\circ}\\ \\ \\ \displaystyle\rm =\frac{1}{4} \times \frac{1}{4}\\ \\ \\ \displaystyle\rm =\frac{1}{16} \\ \\ \\ =\text { RHS } \end{array} \]

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