Chapter 12

$$ \frac{7}{4} \cos \frac{x}{4}=\cos ^{3} \frac{x}{4}+\sin \frac{x}{2} $$

$$ 4 \sin 2 x-\tan ^{2}\left(x-\frac{\pi}{4}\right)=4 $$

$$ (\sin 2 x+\sqrt{3} \cos 2 x)^{2}-5=\cos \left(\frac{\pi}{6}-2 x\right) $$

$$ \cos \frac{4 x}{3}=\cos ^{2} x $$

$$ \sin x+2 \cos x=\cos 2 x-\sin 2 x $$

$$ 32 \cos ^{6} x-\cos 6 x=1 $$

$$ \tan x+\cot x-\cos 4 x=3 $$

$$ 2(1-\sin x-\cos x)+\tan x+\cot x=0 $$

$$ \sin ^{5} x-\cos ^{5} x=\frac{1}{\cos x}-\frac{1}{\sin x} $$

$$ \sin ^{8} 2 x+\cos ^{8} 2 x=\frac{41}{128} $$

$$ \sin ^{10} x+\cos ^{10} x=\frac{29}{64} $$

$$ \sin ^{10} x+\cos ^{10} x=\frac{29}{16} \cos ^{4} 2 x $$

$$ |\cos x|=\cos x-2 \sin x $$

$$ |\cot x|=\cot x+\frac{1}{\sin x} $$

$$ \sqrt{5-2 \sin x}=6 \sin x-1 $$

$$ \sqrt{2+4 \cos x}=\frac{1}{2}+3 \cos x $$

$$ \sqrt{3+2 \tan x-\tan ^{2} x}=\frac{1+3 \tan x}{2} $$

$$ \sqrt{-3 \sin 5 x-\cos ^{2} x-3}+\sin x=1 $$

$$ \tan x+\frac{1}{9} \cot x=\sqrt{\frac{1}{\cos ^{2} x}-1}-1 $$

$$ (1+\cos x) \sqrt{\tan \frac{x}{2}}-2+\sin x=2 \cos x $$

$$ \sqrt{\cos ^{2} x+\frac{1}{2}}+\sqrt{\sin ^{2} x+\frac{1}{2}}=2 $$

$$ \sqrt{1-2 \tan x}-\sqrt{1+2 \cot x}=2 $$

$$ \sqrt{3} \sin x-\sqrt{2 \sin ^{2} x-\sin 2 x+3 \cos ^{2} x}=0 $$

$$ \cos x+\sqrt{\sin ^{2} x-2 \sin 2 x+4 \cos ^{2} x}=0 $$

$$ \sqrt{\cos 2 x}+\sqrt{1+\sin 2 x}=2 \sqrt{\sin x+\cos x} $$

$$ 2 \cot 2 x-3 \cot 3 x=\tan 2 x $$

$$ 6 \tan x+5 \cot 3 x=\tan 2 x $$

$$ \tan \left(x-\frac{\pi}{4}\right) \tan x \tan \left(x+\frac{\pi}{4}\right)=\frac{4 \cos ^{2} x}{\tan \frac{x}{2}-\cot \frac{x}{2}} $$

$$ \sin ^{2} 5 x\left(\sin 7 x \cos x-\sin \frac{x}{2} \cos \frac{3 x}{2}\right)=\frac{\sin \frac{3 x}{2} \cos \frac{x}{2}+\sin x \cos 7 x}{1+\cot ^{2} 5 x} $$

$$ \sin ^{6} x+\sin ^{4} x+\cos ^{6} x+\cos ^{4} x+\sin \frac{x}{2}=3 $$

$$ 1+\cos 2 x \cos 3 x=\frac{1}{2} \sin ^{2} 3 x $$

$$ \sin 5 x+\sin x=2+\cos ^{2} x $$

$$ 3 \sin ^{2} \frac{x}{3}+5 \sin ^{2} x=8 $$

$$ (\sin x+\sqrt{3} \cos x) \sin 3 x=2 $$

$$ 2 \sin \left(\frac{2}{3} x-\frac{\pi}{6}\right)-3 \cos \left(2 x+\frac{\pi}{3}\right)=5 $$

$$ \sin \frac{x}{4}+2 \cos \frac{x-2 \pi}{3}=3 $$

$$ \sin 18 x+\sin 10 x+\sin 2 x=3+\cos ^{2} 2 x $$

$$ \cos 2 x\left(1-\frac{3}{4} \sin ^{2} 2 x\right)=1 $$

$$ \sin x+\cos x=\sqrt{2}+\sin ^{4} 4 x $$

$$ \cos ^{6} 2 x=1+\sin ^{4} x $$

$$ \cot \left(\frac{\pi}{3} \cos 2 \pi x\right)=\sqrt{3} $$

$$ 2 \sin ^{2}\left(\frac{\pi}{2} \cos ^{2} x\right)=1-\cos (\pi \sin 2 x) $$

$$ \text { If } \sin (\pi \cos x)=\cos (\pi \sin x), \text { prove that } \cos \left(x \pm \frac{\pi}{4}\right)=\frac{1}{2 \sqrt{2}} \text { . } $$

If \(\sin (\pi \cot x)=\cos (\pi \tan x)\), prove that either \(\operatorname{cosec} 2 x\) or \(\cot 2 x\) is equal to \(n+\frac{1}{4}\), where \(n\) is a positive or negative integer.

If \(\tan (\pi \cos \theta)=\cot (\pi \sin \theta)\), prove that \(\cos \left(\theta-\frac{\pi}{4}\right)=\pm \frac{1}{2 \sqrt{2}}\).

Find the values of \(\theta\left(0<\theta<360^{\circ}\right)\) satisfying \(\operatorname{cosec} \theta+2=0\).

Find the solution set of \((2 \cos x-1)(3+2 \cos x)=0\) in the interval \(0 \leq x \leq 2 \pi\).

Find the smallest value of \(\theta\) satisfying the equation \(\sqrt{3}(\cot \theta+\tan \theta)=4\).

If \(\cos 20^{\circ}=k\) and \(\cos x=2 k^{2}-1\), then find the possible values of \(x\) between \(0^{\circ}\) and \(360^{\circ}\)

If \(\alpha, \beta\) be unequal values of \(\theta\) satisfying the equation \(a \tan \theta+b \sec \theta=1\), find \(a\) and \(b\) in terms of \(\alpha\) and \(\beta\) and prove that \(\sin \alpha+\cos \alpha+\sin \beta+\cos \beta=\frac{2 b(1-a)}{\left(1+a^{2}\right)}\).