# 6.5 Formulae of sin (A±B), cos (A±B), tan (A±B), sin 2A, cos 2A, tan 2A

6.5 Formulae of sin (A ± B), cos (A ± B), tan (A ± B), sin 2A, cos 2A, tan 2A

(A) Compound Angles Formulae:
$\begin{array}{l}•\text{}\mathrm{sin}\left(A±B\right)=\mathrm{sin}A\mathrm{cos}B±\mathrm{cos}A\mathrm{sin}B\\ •\text{}\mathrm{cos}\left(A±B\right)=\mathrm{cos}A\mathrm{cos}B\mp \mathrm{sin}A\mathrm{sin}B\\ •\text{}\mathrm{tan}\left(A±B\right)=\frac{\mathrm{tan}A±\mathrm{tan}B}{1\mp \mathrm{tan}A\mathrm{tan}B}\end{array}$

(B) Double Angle Formulae:

 sin 2A = 2 sin A cos A cos 2A = cos2 A – sin2 A cos 2A = 2 cos2 A – 1 cos 2A = 1 – 2 sin2 A $\mathrm{tan}2A=\frac{2\mathrm{tan}A}{1-{\mathrm{tan}}^{2}A}$

(C) Half Angle Formulae:

5.5.1 Proving Trigonometric Identities using Addition Formula and Double Angle Formulae

Example 1:
Prove each of the following trigonometric identities.

Solution:
(a)

(b)
$\begin{array}{l}LHS\\ =\frac{\mathrm{cos}\left(A+B\right)}{\mathrm{sin}A\mathrm{cos}B}\\ =\frac{\mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B}{\mathrm{sin}A\mathrm{cos}B}\\ =\frac{\mathrm{cos}A\overline{)\mathrm{cos}B}}{\mathrm{sin}A\overline{)\mathrm{cos}B}}-\frac{\overline{)\mathrm{sin}A}\mathrm{sin}B}{\overline{)\mathrm{sin}A}\mathrm{cos}B}\\ =\frac{\mathrm{cos}A}{\mathrm{sin}A}-\frac{\mathrm{sin}B}{\mathrm{cos}B}\\ =\mathrm{cot}A-\mathrm{tan}B\\ =RHS\text{(proven)}\end{array}$

(c)
$\begin{array}{l}LHS\\ =\mathrm{tan}\left(A+{45}^{o}\right)\\ =\frac{tanA+\mathrm{tan}{45}^{o}}{1-tanA\mathrm{tan}{45}^{o}}\\ =\frac{tanA+1}{1-tanA}←\overline{)\mathrm{tan}{45}^{o}=1}\\ =\frac{\frac{\mathrm{sin}A}{\mathrm{cos}A}+1}{1-\frac{\mathrm{sin}A}{\mathrm{cos}A}}\\ =\frac{\mathrm{sin}A+\mathrm{cos}A}{\overline{)\mathrm{cos}A}}×\frac{\overline{)\mathrm{cos}A}}{\mathrm{cos}A-\mathrm{sin}A}\\ =\frac{\mathrm{sin}A+\mathrm{cos}A}{\mathrm{cos}A-\mathrm{sin}A}\\ =RHS\text{(proven)}\end{array}$