Chapter 12

$$ \text { If } 4 \sin ^{-1} x+\cos ^{-1} x=\pi, \text { then find } x \text { . } $$

$$ \left.\sin x>-\frac{1}{2} \text { \\{Ans. }\left(2 n \pi-\frac{\pi}{6}, 2 n \pi+\frac{7 \pi}{6}\right)\right\\} $$

$$ \left.\cos x<\frac{\sqrt{3}}{2} \text { \\{Ans. }\left(2 n \pi+\frac{\pi}{6}, 2 n \pi+\frac{11 \pi}{6}\right)\right\\} $$

$$ \tan x \geq-\frac{1}{\sqrt{3}} $$

$$ \cot x \leq-1 $$

$$ \sin x<\frac{1}{5} $$

$$ \cos x \geq-0.7 $$

$$ \tan x \leq 5 $$

$$ \cot x>-\frac{\sqrt{3}}{4} $$

$$ \left\\{\begin{array}{l} \sin x<\frac{1}{2} \\ \cos x<\frac{1}{2} \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \sin x>-\frac{\sqrt{3}}{2} \\ \tan x \leq 0 \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \cos x \leq \frac{1}{\sqrt{2}} \\ \cot x>-\sqrt{3} \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \tan x<1 \\ \cot x \geq-\frac{1}{\sqrt{3}} \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \sin x>\frac{1}{5} \\ \cos x<\frac{1}{5} \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \cos x \geq-\frac{3}{5} \\ \tan x<3 \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \sin x<\frac{4}{7} \\ \cot x<2 \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \tan x>0.23 \\ \cot x \leq 0.3 \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \cos x<0 \\ \sin \frac{3}{5} x>0 \end{array}\right\\} $$

$$ \left\\{\begin{array}{l} \sin \frac{x}{2}<\frac{1}{2} \\ \cos 2 x>-\frac{1}{2} \end{array}\right\\} $$

$$ \sin x>\frac{2}{3} ; \cos x<0 $$

$$ \cos x<\frac{1}{2} ; \tan x>-3.5 $$

$$ \sin x<-\frac{\sqrt{3}}{2} ; \cot x \leq 7 $$

$$ \tan x<\frac{1}{\sqrt{3}} ; \cot x<\sqrt{2} $$

$$ \cos x^{2} \geq \frac{1}{2} $$

$$ \sqrt{3} \sin 2 x+\cos 2 x<1 $$

$$ \cos 3 x+\sqrt{3} \sin 3 x<-\sqrt{2} $$

$$ \cos 2 x+\cos x>0 $$

$$ \frac{\cos x}{1+\cos 2 x}<0 $$

$$ \sin 3 x>\cos 3 x $$

$$ \tan x+3 \cot x-4>0 $$

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

$$ 2 \sin ^{2} \frac{x}{2}+\cos 2 x<0 $$

$$ \left.\tan ^{3} x+3>3 \tan x+\tan ^{2} x\right\\} $$

$$ \sin \theta-\cos \theta<0 $$

$$ \frac{\sin 3 x-\cos 3 x}{\sin 3 x+\cos 3 x}<0 $$

$$ 5 \sin ^{2} x-3 \sin x \cos x-36 \cos ^{2} x>0 $$

$$ 2 \sin ^{2} x-4 \sin x \cos x+9 \cos ^{2} x>0 $$

$$ \cos ^{2} x+3 \sin ^{2} x+2 \sqrt{3} \sin x \cos x<1 $$

$$ 3 \sin ^{2} x+\sin 2 x-\cos ^{2} x \geq 2 $$

$$ \sqrt{3} \cos ^{-2} x<4 \tan x $$

$$ \sin 4 x+\cos 4 x \cot 2 x>1 $$

$$ 2+\tan 2 x+\cot 2 x<0 $$

$$ 2 \cos x(\cos x-\sqrt{8} \tan x)<5 $$

$$ \sin x+\cos x<\frac{1}{\cos x} $$

$$ \sin ^{6} x+\cos ^{6} x<\frac{7}{16} $$

$$ \cot x+\frac{\sin x}{\cos x-2} \geq 0 $$

$$ \cos ^{2} 2 x+\cos ^{2} x \leq 1 $$

$$ 8 \sin ^{2} \frac{x}{2}+3 \sin x-4>0 $$

$$ \sin x+\cos x>\sqrt{2} \cos 2 x $$

$$ \tan x+\tan 2 x+\tan 3 x>0 $$