JEE Main & Advanced Mathematics Trigonometric Identities Conditional Trigonometrical Identities

We have certain trigonometric identities.

Like, \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]  and  \[1+{{\tan }^{2}}\theta ={{\sec }^{2}}\theta \] etc.

Such identities are identities in the sense that they hold for all value of the angles which satisfy the given condition among them and they are called conditional identities.

(1) If \[A+B+C={{180}^{o}}\], then

(i) \[\sin 2A+\sin 2B+\sin 2C=4\sin A\sin B\sin C\]

(ii) \[\sin 2A+\sin 2B-\sin 2C=4\cos A\cos B\sin C\]

(iii) \[\sin (B+C-A)+\sin (C+A-B)+\sin (A+B-C)\]\[=4\sin A\sin B\sin C\]

(iv) \[\cos 2A+\cos 2B+\cos 2C=-1-4\cos A\cos B\cos C\]

(v) \[\cos 2A+\cos 2B-\cos 2C=1-4\sin A\sin B\cos C\]

(2) If \[A+B+C={{180}^{o}}\], then

(i) \[\sin A+\sin B+\sin C=4\cos \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}\]

(ii) \[\sin A+\sin B-\sin C=4\sin \frac{A}{2}\sin \frac{B}{2}\cos \frac{C}{2}\]

(iii) \[\cos A+\cos B+\cos C=1+4\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\]

(iv) \[\cos A+\cos B-\cos C=-1+4\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}\]

(v) \[\frac{\cos A}{\sin B\sin C}+\frac{\cos B}{\sin C\sin A}+\frac{\cos C}{\sin A\sin B}=2\]

(3) If \[A+B+C=\pi \], then

(i) \[{{\sin }^{2}}A+{{\sin }^{2}}B-{{\sin }^{2}}C=2\sin A\sin B\cos C\]

(ii) \[{{\cos }^{2}}A+{{\cos }^{2}}B+{{\cos }^{2}}C=1-2\cos A\cos B\cos C\]

(iii) \[{{\sin }^{2}}A+{{\sin }^{2}}B+{{\sin }^{2}}C=1-2\sin A\sin B\cos C\]

(4) If \[A+B+C=\pi ,\] then

(i) \[{{\sin }^{2}}\frac{A}{2}+{{\sin }^{2}}\frac{B}{2}+{{\sin }^{2}}\frac{C}{2}=1-2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\]

(ii) \[{{\cos }^{2}}\frac{A}{2}+{{\cos }^{2}}\frac{B}{2}+{{\cos }^{2}}\frac{C}{2}=2+2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\]

(iii) \[{{\sin }^{2}}\frac{A}{2}+{{\sin }^{2}}\frac{B}{2}-{{\sin }^{2}}\frac{C}{2}=1-2\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}\]

(iv) \[{{\cos }^{2}}\frac{A}{2}+{{\cos }^{2}}\frac{B}{2}-{{\cos }^{2}}\frac{C}{2}=2\cos \frac{A}{2}\cos \frac{B}{2}\sin \frac{C}{2}\]

(5) If \[x+y+z=\frac{\pi }{2}\], then

(i) \[{{\sin }^{2}}x+{{\sin }^{2}}y+{{\sin }^{2}}z=1-2\sin x\sin y\sin z\]

(ii) \[{{\cos }^{2}}x+{{\cos }^{2}}y+{{\cos }^{2}}z=2+2\sin x\sin y\sin z\]

(iii) \[\sin 2x+\sin 2y+\sin 2z=4\cos x\cos y\cos z\]

(6) If \[A+B+C=\pi \], then

(i) \[\tan A+\tan B+\tan C=\tan A\tan B\tan C\]

(ii) \[\cot B\cot C+\cot C\cot A+\cot A\cot B=1\]

(iii) \[\tan \frac{B}{2}\tan \frac{C}{2}+\tan \frac{C}{2}\tan \frac{A}{2}+\tan \frac{A}{2}\tan \frac{B}{2}=1\]

(iv) \[\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\cot \frac{A}{2}\cot \frac{B}{2}\cot \frac{C}{2}\]