Trigonometric Ratios of Allied Angles

Q.

sec $(\pi -\theta )$ = -sec$\theta$

1. True

2. False

Q. If $A,B$ are supplementary angles, then the value of $\sin A\sin B-\cos A\cos B$ is

Q. The value of $\text{cosec}\theta +\sec (270°-\theta )-\text{cosec}(270°+\theta )+\sec (180°-\theta )$ is

1. $0$
2. $1$
3. $2(\sec \theta +\text{cosec}\theta )$
4. $2\sec \theta$

Q. If $\sin x=\frac{12}{13}$ and $x\in [0,\pi ]$, then the value(s) of $\sec x+\tan x$ is/are

1. $5$
2. $4$
3. $-5$
4. $-4$

Q. Fundamental period of the function $f\left(x\right)=|\sin \pi x|$ is

1. $1$
2. $2$
3. ${\pi }^2$
4. $2\pi$

Q. $\frac{\cos (90°+\theta )\text{cosec}(270°+\theta )\cos (180°+\theta )}{\sec (-\theta )\sin (270°+\theta )\sin (360°-\theta )}=$

1. $-1$
2. $1$
3. ${\cot }^2\theta$
4. $\tan \theta$

Q. If $ABCD$ (in order) is a quadrilateral inscribed in a circle, then which of the following is/are always true?

1. $\sec B=\sec D$
2. $\cos A+\cos C=0$
3. $\text{cosec}A=\text{cosec}C$
4. $\tan B+\tan D=0$

Q. The value of
${\sin }^2{5}^{\circ }+{\sin }^2{10}^{\circ }+{\sin }^2{15}^{\circ }+{\sin }^2{75}^{\circ }+{\sin }^2{80}^{\circ }+{\sin }^2{85}^{\circ }+{\sin }^2{90}^{\circ }\text{is}$

Q. The value of the expression $\tan \frac{\pi }{7}+2\tan \frac{2\pi }{7}+4\tan \frac{4\pi }{7}+8\cot \frac{8\pi }{7}$ is equal to

1. $\text{cosec}\frac{2\pi }{7}+\cot \frac{2\pi }{7}$
2. $\tan \frac{\pi }{14}-\cot \frac{\pi }{14}$
3. $\frac{\sin \frac{2\pi }{7}}{1-\cos \frac{2\pi }{7}}$
4. $\frac{1+\cos \frac{\pi }{7}+\cos \frac{2\pi }{7}}{\sin \frac{\pi }{7}+\sin \frac{2\pi }{7}}$

Q. The value of $\sin {765}^{\circ }+\text{cosec}(-{1110}^{\circ })-\sin {405}^{\circ }+\cot {585}^{\circ }$ is equal to

1. $1$
2. $-1$
3. $2$
4. $-2$

Q. $\frac{\sin 18°}{\cos 72°}\times \frac{\tan 150°}{cot60°}$ is equal to

1. 
   -$\sqrt{3}$
2. 
   -$1$
3. 
   $\frac{1}{2}$
4. 
   $\frac{1}{\sqrt{2}}$

Q. If $A,B,C,D$ are the angles of a cyclic quadrilateral, then $\cos A+\cos B+\cos C+\cos D$ is equal to

1. $-1$
2. $0$
3. $-2$
4. $1$

Q. The value of $\frac{\cos 15\pi +\tan \left(\frac{5\pi }{4}\right)}{\sin 16\pi +\sec 6\pi }$ is

1. $0$
2. $1$
3. $\frac{1}{2}$
4. $-1$

Q. If $f\left(x\right)=\frac{1+\sin ({90}^{\circ }+2x)-\cos ({270}^{\circ }+2x)}{\cos ({360}^{\circ }+4x)}$ then value of $\tan {30}^{\circ }f\left({30}^{\circ }\right)+\tan {45}^{\circ }f\left({45}^{\circ }\right)$ is

1. $0$
2. $1-\sqrt{3}$
3. $3\sqrt{3}-1$
4. $\sqrt{3}-1$

Q. The value of $\frac{\sin \theta -\sin 2\theta }{1-\cos \theta +\cos 2\theta }$ is

1. $\tan \frac{\theta }{2}$
2. $\cot \frac{\theta }{2}$
3. $\tan \theta$
4. $-\tan \theta$

Q. The value of $\tan \left(\frac{\pi }{20}\right)\tan \left(\frac{3\pi }{20}\right)\tan \left(\frac{5\pi }{20}\right)\tan \left(\frac{7\pi }{20}\right)\tan \left(\frac{9\pi }{20}\right)$ is

1. $0$
2. $1$
3. $-1$
4. $\infty$

Q. $\sec \left(\frac{3\pi }{2}-\theta \right)\sec \left(\theta -\frac{5\pi }{2}\right)+\tan \left(\frac{5\pi }{2}+\theta \right)\tan \left(\theta -\frac{3\pi }{2}\right)$
is equal to

1. 
   0
2. 
   -1
3. 
   1
4. 
   2

Q.

If one root of the equation $a{x}^2+x-3=0$ is $-1$, then what is the other root?

1. $\frac{1}{4}$

2. $\frac{1}{2}$

3. $\frac{3}{4}$

4. $1$

Q. The value of $\cos (\frac{\pi }{4}+A)\cos (\frac{\pi }{4}-B)-\sin (\frac{\pi }{4}+A)\sin (\frac{\pi }{4}-B)$ is

1. $\cos (A+B)$
2. $\cos (A-B)$
3. $\sin (A+B)$
4. $\sin (B-A)$

Q. The value of $4{\cos }^3{40}^{\circ }-3\sin {50}^{\circ }$ is

1. $-\frac{\sqrt{3}}{2}$
2. $\frac{1}{\sqrt{2}}$
3. $\frac{1}{2}$
4. $-\frac{1}{2}$

Q. In $△ABC$, if $\cot x=\cot A+\cot B+\cot C$, then $\sin (A-x)\sin (B-x)\sin (C-x)$ is

1. ${\sin }^{3}x$
2. ${\cos }^{3}x$
3. ${\sin }^{2}x$
4. $3\sin x$

Q. If $5\cos \theta \sin \theta =1,$ then the value of ${\tan }^3\theta +{\cot }^3\theta$ is

1. $110$
2. $115$
3. $125$
4. $100$