# JEE Main & Advanced Mathematics Trigonometrical Ratios and Identities Trigonometric Ratio of Multiple of an Angle

## Trigonometric Ratio of Multiple of an Angle

Category : JEE Main & Advanced

(1) $\sin 2A=2\sin A\cos A$$=\frac{2\tan A}{1+{{\tan }^{2}}A}$

(2) $\frac{\sqrt{5}-1}{4}$$\frac{1}{4}\sqrt{10-2\sqrt{5}}$

$={{\cos }^{2}}A-{{\sin }^{2}}A$$2-\sqrt{3}$; where $A\ne (2n+1)\frac{\pi }{4}$.

(3) $\tan 2A=\frac{2\tan A}{1-{{\tan }^{2}}A}$

(4) $\sin 3A=3\sin A-4{{\sin }^{3}}A$$=4\sin ({{60}^{o}}-A).\sin A.\sin ({{60}^{o}}+A)$

(5) $\cos 3A=4{{\cos }^{3}}A-3\cos A$$=4\cos ({{60}^{o}}-A).\cos A.\cos ({{60}^{o}}+A)$

(6) $\tan 3A=\frac{3\tan A-{{\tan }^{3}}A}{1-3{{\tan }^{2}}A}=\tan ({{60}^{o}}-A).\tan A.\tan ({{60}^{o}}+A)$, where $A\ne n\pi +\pi /6$

(7) $\sin 4\theta =4\sin \theta .{{\cos }^{3}}\theta -4\cos \theta {{\sin }^{3}}\theta$

(8) $\cos 4\theta =8{{\cos }^{4}}\theta -8{{\cos }^{2}}\theta +1$

(9) $\tan 4\theta =\frac{4\tan \theta -4{{\tan }^{3}}\theta }{1-6{{\tan }^{2}}\theta +{{\tan }^{4}}\theta }$

(10) $\sin 5A=16{{\sin }^{5}}A-20{{\sin }^{3}}A+5\sin A$

(11) $\cos 5A=16{{\cos }^{5}}A-20{{\cos }^{3}}A+5\cos A$

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