JEE Main & Advanced Mathematics Trigonometric Identities Trigonometric Ratio of Multiple of an Angle

(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) \[\cos 5A=16{{\cos }^{5}}A-20{{\cos }^{3}}A+5\cos A\]