Trigonometric Functions of Sum and Difference of Three Angles

(i) sin(A + B + C) = sin A cos B cos C + cos A sin B cos C + cos A cos B sin C − sin A sin B sin C

or

sin(A + B + C) = cos A cos B cos C (tan A + tan B + tan C − tan A tan B tan C)

(ii) cos(A + B + C) = cos A cos B cos C − sin A sin B cos C − sin A cos B sin C − cos A sin B sin C

or

cos(A + B + C) = cos A cos B cos C (1 − tan A tan B − tan B tan C − tan C tan A)

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

(iv) cot(A + B + C) \[=\frac{\cot A\cot B\cot C-\cot A-\cot B-\cot C}{\cot A\cot B+\cot B\cot C+\cot C\cot A-1}\]

If (A + B + C = 180°):

1. sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C

2. cos 2A + cos 2B + cos 2C = −1 − 4 cos A cos B cos C

3. cos 2A + cos 2B − cos 2C = 1 − 4 sin A sin B cos C

4. \[\sin A+\sin B+\sin C=4\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}\]

5. \[\cos A+\cos B+\cos C=1+4\sin\frac{A}{2}\sin\frac{B}{2}\sin\frac{C}{2}\]

6. cos A + cos B − cos C = \[-1+4\cos\frac{A}{2}\cos\frac{B}{2}\sin\frac{C}{2}\]

7. tan A + tan B + tan C = tan A tan B tan C

8. cot A cot B + cot B cot C + cot C cot A = 1

9. \[\tan\frac{A}{2}\tan\frac{B}{2}+\tan\frac{B}{2}\tan\frac{C}{2}+\tan\frac{C}{2}\tan\frac{A}{2}=1\]

10. \[\cot\frac{A}{2}+\cot\frac{B}{2}+\cot\frac{C}{2}=\cot\frac{A}{2}\cot\frac{B}{2}\cot\frac{C}{2}\]