Tangent, Cotangent, Secant, Cosecant in Terms of Sine and Cosine

Q. The value of ${\cos }^{-1}\left(\cos \frac{13\pi }{6}\right)$ is

1. $\frac{13\pi }{6}$
2. $\frac{7\pi }{6}$
3. $\frac{5\pi }{6}$
4. $\frac{\pi }{6}$

Q. The value of $\sin \frac{\pi }{5}\sin \frac{2\pi }{5}\sin \frac{3\pi }{5}\sin \frac{4\pi }{5}$ is

1. $\frac{5}{16}$
2. $\frac{3}{16}$
3. $\frac{11}{16}$
4. $\frac{13}{16}$

Q. The value of $\sin {75}^{\circ }+\cos {75}^{\circ }$ is

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

Q. If $\frac{sin(x+y)}{sin(x-y)}=\frac{a+b}{a-b},$ Show that $\frac{tanx}{tany}=\frac{a}{b}.$

Q. The expression $\frac{\tan A}{1-\cot A}+\frac{\cot A}{1-\tan A}$ can be written as :

1. $\sin A\cdot \cos A+1$
2. $\sec A\cdot \text{cosec}A+1$
3. $\tan A+\cot A$
4. $\sec A+\text{cosec}A$

Q.

$\frac{tanA}{1-cotA}+\frac{cotA}{1-tanA}=(secAcosecA+1)$

Q. If $\sqrt{2}\sin A=\sin B-{\sin }^{3}B$ and $\sqrt{2}\cos A=\cos B+{\cos }^{3}B$, then the possible value(s) of $\sin (A-B)$ is/are

1. $\sin (A-B)=-\frac{1}{2\sqrt{2}}\sin 2B$
2. $\sin (A-B)=\frac{1}{2\sqrt{2}}\sin 2B$
3. $\sin (A-B)=\frac{1}{3}$
4. $\sin (A-B)=-\frac{1}{3}$

Q. The value of $\cos (n+1)\alpha \cos (n-1)\alpha +\sin (n+1)\alpha \sin (n-1)\alpha$ is

1. $\cos 2n\alpha$
2. $\cos 2\alpha$
3. $\sin 2n\alpha$
4. $\sin 2\alpha$

Q. If $\sin A=\frac{12}{13},\cos B=-\frac{3}{5},0<A<\frac{\pi }{2},$ $\pi <B<\frac{3\pi }{2}$, then the value of $\sin (A+B)$ is

1. $\frac{33}{65}$
2. $-\frac{1}{63}$
3. $-\frac{56}{65}$
4. $-\frac{63}{65}$

Q. If $\frac{1+\sin 2x}{1-\sin 2x}={\cot }^2(a+x)$$\forall x\in R-(n\pi +\frac{\pi }{4}),$ $n\in N$ then the possible value of $a$ is

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

Q. If $\sin A\sin B-\cos A\cos B+1=0$, then the value of $1-\cot A\tan B$ is

Q. If $A+B+C=0,$ then value of $\left|\begin{array}{ccc}1& \cos C& \cos B\\ \cos C& 1& \cos A\\ \cos B& \cos A& 1\end{array}\right|$ is

Q.

If $\theta \in \left(0,\frac{\mathrm{\pi }}{4}\right)$ and $t_1={\left(\tan \theta \right)}^{\tan \theta },t_2={\left(\tan \theta \right)}^{cot\theta },t_3={\left(cot\theta \right)}^{\tan \theta }$ and $t_4={\left(cot\theta \right)}^{\tan \theta },$ then

1. $t_1>t_2>t_3>t_4$

2. $t_4>t_3>t_1>t_2$

3. $t_3>t_1>t_2>t_4$

4. $t_2>t_3>t_1>t_4$

Q. The value of $\frac{\cos {15}^{\circ }+\sin {15}^{\circ }}{\cos {15}^{\circ }-\sin {15}^{\circ }}$ is

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

Q. The value of $(\cos {75}^{\circ }-\cos {15}^{\circ }{)}^2+(\sin {75}^{\circ }-\sin {15}^{\circ }{)}^2$ is

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

Q. The expression $\sin A(1+\tan A)+\cos A(1+\cot A)$ is equivalent to

1. $\sin A+\cos A$
2. $\sec A+\tan A$
3. $\text{cosec}A+\cot A$
4. $\text{cosec}A+\sec A$

Q. The angle (in degree) between the hour hand and the minute hand in a circular clock at $03:25$ hours is

1. ${67.5}^{\circ }$
2. ${75}^{\circ }$
3. ${47.5}^{\circ }$
4. ${52.5}^{\circ }$

Q. If $\sin (\theta +\alpha )=\cos (\theta +\alpha )$, then which of the following is/are correct?
where $\theta ,\alpha \in (0,\frac{\pi }{2})-\left\{\frac{\pi }{4}\right\}$

1. $\tan \alpha =\frac{1-\tan \theta }{1+\tan \theta }$
2. $\tan \alpha =\frac{1+\tan \theta }{1-\tan \theta }$
3. $\tan \theta =\frac{1+\tan \alpha }{1-\tan \alpha }$
4. $\tan \theta =\frac{1-\tan \alpha }{1+\tan \alpha }$

Q. If $\frac{tanA}{1-cotA}+\frac{cotA}{1-tanA}=1+secA.cosecA.$

Q. If $\sin A+\sin B+\sin C=0$ and $\cos A+\cos B+\cos C=0$, then the value of $\sin \left(\frac{A-B}{2}\right)$ is
$(\text{where}A,B,C\in [0,2\pi ])$

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

Q. The numerical value of $(1-\cos x)(1+\cos x)(1+{\cot }^{2}x)$ is

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

Q. The angle between the minute hand and the hour hand of a clock when the time is $7:20\text{a.m.}$ is

1. ${90}^{\circ }$
2. ${100}^{\circ }$
3. ${75}^{\circ }$
4. ${120}^{\circ }$

Q. If ${\cos }^{6}A+{\sin }^{6}A=1-k{\sin }^{2}2A$, then the value of $4k$ is

Q. If $\cot \frac{2x}{3}+\tan \frac{x}{3}=\text{cosec}\frac{kx}{3}$, then the value of $k$ is

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

Q.

Prove: $(\text{cosec}A-\sin A)(\sec A-\cos A)=\frac{1}{\tan A+\cot A}$

Q. Which of the following is/are correct regarding their fundamental period?

1. $\frac{1}{|\sin x|+|\cos x|}\to \pi /2$
2. $|\sin x+\cos x|\to \pi$
3. $|\sin x+\cos x|+|\sin x-\cos x|\to 3\pi /2$
4. $\frac{|\sin x|}{\cos x}+\frac{\sin x}{|\cos x|}\to 2\pi$

Q. Let $A(5,12),B(-13cos\theta ,13sin\theta )$ and $C(13sin\theta ,-13cos\theta )$ be vertices of $\Delta ABC$ where $\theta ϵR$. The locus of orthocenter of $\Delta ABC$ is

1. x – y + 7 = 0
2. x – y - 7 = 0
3. x + y - 7 = 0
4. x + y + 7 = 0

Q. Which of the following is/are irrational numbers ?

1. $\sin {15}^{\circ }$
2. $\cos {15}^{\circ }$
3. $\sin {15}^{\circ }\cos {15}^{\circ }$
4. $\sec {15}^{\circ }+\text{cosec}{15}^{\circ }$

Q. If $x\cos \theta =y\cos (\theta +\frac{\pi }{3})=z\sin (\theta -\frac{\pi }{6})$, where $y\ne 0$, then which of the following is/are correct?

1. $y=z\forall \theta \in R$
2. $y+z=0\forall \theta \in R$
3. If $x=0$, then $\theta$ lies in the first and third quadrant.
4. There is only one value of $x$ satisfying the equation $x+y+z=0$

Q. The value of ${\sin }^{2}A+{\sin }^2(A+B)-2\sin A\cos B\sin (A+B)$ when $B={45}^{\circ }$ is

1. $\frac{1}{\sqrt{2}}$
2. $\frac{1}{2}$
3. ${\sin }^{2}A+\frac{1}{\sqrt{2}}$
4. ${\sin }^{2}A+\frac{1}{2}$