Prove the identity $$$\left(\frac{1}{\cos x}-\tan x\right)\left(\frac{1}{\sin x}+1\right) \equiv \frac{1}{\tan x}$$$.  Hence solve the equation $$$\left(\frac{1}{\cos x}-\tan x\right)\left(\frac{1}{\sin x}+1\right)=2 \tan ^{2} x$$$ for $$$0^{\circ} \leqslant x \leqslant 180^{\circ}$$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................

Mathematics

IGCSE&ALevel

CAIE

Exam No：9709_w20_qp_12 Year：2020 Question No：6(b)

Answer:

Uses (a) $\rightarrow \frac{1}{\tan x}=2 \tan ^{2} x \tan ^{3} x=\frac{1}{2}$
$(x=) 38.4^{\circ}$

1.5.5 find all the solutions of simple trigonometrical equations lying in a specified interval (general forms of solution are not included).