5.5.1 Proving Trigonometric Identities using Addition Formula and Double Angle Formulae (Examples)

Example 2:
Prove each of the following trigonometric identities.

Solution:
(a)

(b)
$\begin{array}{l}RHS\\ =\mathrm{cot}A+\mathrm{tan}2A\\ =\frac{\mathrm{cos}A}{\mathrm{sin}A}+\frac{\mathrm{sin}2A}{\mathrm{cos}2A}\\ =\frac{\mathrm{cos}A\mathrm{cos}2A+\mathrm{sin}A\mathrm{sin}2A}{\mathrm{sin}A\mathrm{cos}2A}\\ =\frac{\mathrm{cos}A\left({\mathrm{cos}}^{2}A-{\mathrm{sin}}^{2}A\right)+\mathrm{sin}A\left(2\mathrm{sin}A\mathrm{cos}A\right)}{\mathrm{sin}A\mathrm{cos}2A}\\ =\frac{{\mathrm{cos}}^{3}A-\mathrm{cos}A{\mathrm{sin}}^{2}A+2{\mathrm{sin}}^{2}A\mathrm{cos}A}{\mathrm{sin}A\mathrm{cos}2A}\\ =\frac{{\mathrm{cos}}^{3}A+\mathrm{cos}A{\mathrm{sin}}^{2}A}{\mathrm{sin}A\mathrm{cos}2A}\\ =\frac{\mathrm{cos}A\left({\mathrm{cos}}^{2}A+{\mathrm{sin}}^{2}A\right)}{\mathrm{sin}A\mathrm{cos}2A}\\ =\frac{\mathrm{cos}A}{\mathrm{sin}A\mathrm{cos}2A}←\overline{){\mathrm{sin}}^{2}A+{\mathrm{cos}}^{2}A=1}\\ =\left(\frac{\mathrm{cos}A}{\mathrm{sin}A}\right)\left(\frac{1}{\mathrm{cos}2A}\right)\\ =\mathrm{cot}A\mathrm{sec}2A\end{array}$
(c)

Example 3:
(a) Given that  such that P is an acute angle and Q is an obtuse angle, without using tables or a calculator, find the value of cos (P + Q).
(b) Given that  such that A and B are angles in the third and fourth quadrants respectively, without using tables or a calculator, find the value of sin (A  B).

Solution:
(a)

(b)