Trigonometric Ratios of Multiples of an Angle

Q. If $\left[\cos |\alpha |\right]=\left\{\sin x\right\},\alpha \in R,$ where $\{.\}$ is the fractional part function and $[.]$ is the greatest integer function, then the value of $5-\log \tan |\frac{\pi }{4}-x|$ is

Q. If $x+\frac{1}{x}=2\cos \theta ,$ then find the value of $x^3+\frac{1}{x^3}.$

1. $cos3\theta$
2. $2cos3\theta$
3. $\frac{1}{2}cos3\theta$
4. $\frac{1}{3}cos3\theta$

Q. Let $f\left(x\right)=\frac{1-\sin 2x+\cos 2x}{2\cos 2x}$ and $\alpha +\beta =\frac{5\pi }{4}.$ Then the value of $f\left(\alpha \right)\cdot f\left(\beta \right)$ is

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

Q. The value of ${\cos }^3\frac{\pi }{8}\cos \frac{3\pi }{8}+{\sin }^3\frac{\pi }{8}\sin \frac{3\pi }{8}$ is :

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

Q. If $0<{\cos }^{-1}x<1$ and $1+\sin \left({\cos }^{-1}x\right)+{\sin }^2\left({\cos }^{-1}x\right)+{\sin }^3\left({\cos }^{-1}x\right)+\dots \infty =2$, then the value of $12x^2$ is

Q. $Ifx⩾1orx⩽-1,thencose{c}^{-1}\left(x\right)=$

1. $\frac{1}{si{n}^{-1}\left(x\right)}$
2. $se{c}^{-1}(\frac{\pi }{2}-x)$
3. $si{n}^{-1}\left(\frac{1}{x}\right)$
4. None of these

Q. The domain of ${\sin }^{-1}\left(3x\right)$ is

1. $\left(-1,1\right]$
2. $\left(-3,3\right]$
3. $\left(-3,3\right)$
4. $\left(\frac{-1}{3},\frac{1}{3}\right]$
   

Q. $Let\alpha ,\beta besuchthat\pi <\alpha -\beta <3\pi .Ifsin\alpha +sin\beta =\frac{21}{65}andcos\alpha +cos\beta =-\frac{27}{65},thenthevalueofcos\frac{\alpha -\beta }{2}is$

1. $\frac{-6}{65}$
2. $\frac{3}{\sqrt{130}}$
3. $\frac{6}{65}$
4. $-\frac{3}{\sqrt{130}}$

Q. The value of $\sin \frac{\pi }{n}+\sin \frac{3\pi }{n}+\sin \frac{5\pi }{n}+\cdots \text{to}n\text{terms}$ is

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

Q.

For a positive integer n, let $f_n\left(\theta \right)=\left(tan\frac{\theta }{2}\right)(1+sec\theta )(1+sec2\theta )(1+sec4\theta )......(1+sec{2}^n\theta ).Then$

1. $f_2\left(\frac{\pi }{16}\right)=1$

2. $f_3\left(\frac{\pi }{32}\right)=1$

3. $f_4\left(\frac{\pi }{64}\right)=1$

4. $f_5\left(\frac{\pi }{128}\right)=1$

Q. For all permissible values of $A,2A$, following holds true.
$\left(i\right)\cot A+\tan A=\frac{1}{\sin A\cos A}=2\text{cosec}2A\left(ii\right)\cot A-\tan A=\frac{{\cos }^{2}A-{\sin }^{2}A}{\sin A\cos A}=2\cot 2A\left(iii\right)2\cot A=2(\text{cosec}2A+\cot 2A)\Rightarrow \text{cosec}2A+\cot 2A=\cot A$

Also to evaluate a series of form $f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)$ when $f\left(x\right)$ can be expressed as $g\left(x\right)-g\left(2x\right)$, we can use the following technique,
$f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)=\left(g\right(x)-g(2x\left)\right)+\left(g\right(2x)-g(4x\left)\right)+\cdots \left(g\right(2^{n}x)-g(2^{n+1}x\left)\right)=g\left(x\right)-g\left(2^{n+1}x\right)$

Based on the above information, solve the following questions for all permissible values of $x$.

The value of $\text{cosec}2x+\text{cosec}4x+\text{cosec}8x+\text{cosec}16x+\text{cosec}32x$

1. $\cot x-\cot 64x$
2. $\cot 2x-\cot 64x$
3. $\cot x-\cot 32x$
4. $\cot 2x-\cot 32x$

Q. The value of x for which $sin\left(co{t}^{-1}\right(1+x\left)\right)=cos\left(ta{n}^{-1}x\right)$ is:

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

Q. If $sin6\theta =32co{s}^5\theta sin\theta -32co{s}^3\theta sin\theta +3x,$ then x=

1. $cos\theta$
2. $cos2\theta$
3. $sin\theta$
4. $sin2\theta$

Q. $\mathrm{If}\sin \theta =0.8\mathrm{and}cot\theta <0,\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\sin 2\theta \mathrm{is}$.

1. 0.64
2. 0.96
3. -0.96
4. -0.64

Q. Which of the following statements is (are) CORRECT?
($Q$ denotes the set of rational numbers)

1. If $\cos 2\theta \in Q$ and $\sin 2\theta \in Q,$ then $\tan \theta \in Q$ (if defined)
2. If $\tan \theta \in Q,$ then $\sin 2\theta ,\cos 2\theta$ and $\tan 2\theta \in Q$ (if defined)
3. If $\sin \theta \in Q$ and $\cos \theta \in Q,$ then $\tan 3\theta \in Q$ (if defined)
4. If $\sin \theta \in Q,$ then $\cos 3\theta \in Q$

Q. The value of ${\cos }^3\frac{\pi }{8}\cos \frac{3\pi }{8}+{\sin }^3\frac{\pi }{8}\sin \frac{3\pi }{8}$ is :

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

Q. If $\cos x+\sin x=\frac{1}{2}$, where $x\in (0,\pi )$, then the maximum possible value of $\tan x$ is

1. $\frac{-4-\sqrt{7}}{3}$
2. $\frac{4+\sqrt{7}}{3}$
3. $\frac{-4+\sqrt{7}}{3}$
4. $\frac{4-\sqrt{7}}{3}$

Q.

Prove the identity ${\left(sec\theta -\tan \theta \right)}^2=\frac{1-\sin \theta }{1+\sin \theta }$

Q.

$Ifsi{n}^{3}xsin3x={\sum }_{m=0}^n{c}_{m}cosmxwhere{c}_0,c_1,c_2.\cdots \cdots ,c_{n}areconstantsand{c}_n\ne 0,thenthevalueofnis$

1. 15

2. 6

3. 1

4. 0

Q. If $2sec2\alpha =tan\beta +cot\beta ,$ then one of the values of
$\alpha +\beta$ is

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

Q. $si{n}^4\frac{\pi }{8}+si{n}^4\frac{3\pi }{8}+si{n}^4\frac{5\pi }{8}+si{n}^4\frac{7\pi }{8}=$

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

Q. Which of the following statements is (are) CORRECT?
($Q$ denotes the set of rational numbers)

1. If $\cos 2\theta \in Q$ and $\sin 2\theta \in Q,$ then $\tan \theta \in Q$ (if defined)
2. If $\tan \theta \in Q,$ then $\sin 2\theta ,\cos 2\theta$ and $\tan 2\theta \in Q$ (if defined)
3. If $\sin \theta \in Q$ and $\cos \theta \in Q,$ then $\tan 3\theta \in Q$ (if defined)
4. If $\sin \theta \in Q,$ then $\cos 3\theta \in Q$

Q. If $L=\left(4\tan 1^{\circ }\right)\left(4\tan 2^{\circ }\right)\left(4\tan 3^{\circ }\right)\dots \left(4\tan {10}^{\circ }\right),$$M=\left(\tan 2^{\circ }\right)\left(\tan 4^{\circ }\right)\left(\tan 6^{\circ }\right)\dots \left(\tan {20}^{\circ }\right),$ and $N=(1+\tan 1^{\circ })(1-\tan 1^{\circ })(1+\tan 2^{\circ })(1-\tan 2^{\circ })\dots (1+\tan {10}^{\circ })(1-\tan {10}^{\circ })$, then the value of $\frac{L}{128MN}$ is

Q. If $sin6\theta =32co{s}^5\theta sin\theta -32co{s}^3\theta sin\theta +3x,$ then x=

1. $cos\theta$
2. $cos2\theta$
3. $sin\theta$
4. $sin2\theta$

Q. Let $f\left(\theta \right)=\sin \theta (\sin \theta +\sin 3\theta ),$ then:

1. $\ge 0onlywhen\theta \ge 0$
2. $\le 0forallral\theta$
3. $\ge 0forallreal\theta$
4. $\le 0onlywhen\theta \le 0$

Q. For all permissible values of $A,2A$, following holds true.
$\left(i\right)\cot A+\tan A=\frac{1}{\sin A\cos A}=2\text{cosec}2A\left(ii\right)\cot A-\tan A=\frac{{\cos }^{2}A-{\sin }^{2}A}{\sin A\cos A}=2\cot 2A\left(iii\right)2\cot A=2(\text{cosec}2A+\cot 2A)\Rightarrow \text{cosec}2A+\cot 2A=\cot A$

Also to evaluate a series of form $f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)$ when $f\left(x\right)$ can be expressed as $g\left(x\right)-g\left(2x\right)$, we can use the following technique,
$f\left(x\right)+f\left(2x\right)+f\left(4x\right)+\cdots +f\left(2^{n}x\right)=\left(g\right(x)-g(2x\left)\right)+\left(g\right(2x)-g(4x\left)\right)+\cdots \left(g\right(2^{n}x)-g(2^{n+1}x\left)\right)=g\left(x\right)-g\left(2^{n+1}x\right)$

Based on the above information, solve the following questions for all permissible values of $x$.

The value of $\tan x+\frac{1}{2}\tan \frac{x}{2}+\frac{1}{2^2}\tan \frac{x}{2^2}+\cdots +\frac{1}{2^{10}}\tan \frac{x}{2^{10}}$ is

1. $\frac{1}{2^{11}}\cot \frac{x}{2^{11}}-2\cot 2x$
2. $\frac{1}{2^{10}}\cot \frac{x}{2^{10}}-2\cot 2x$
3. $\frac{1}{2^{10}}\cot \frac{x}{2^{10}}-2\cot x$
4. None of the above

Q. The value of ${\cos }^3({60}^{\circ }-A)-{\cos }^3({60}^{\circ }+A)$ is

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

Q. If $y={\cos }^2({45}^{\circ }+x)+(\sin x-\cos x{)}^2$, where $x\in (0,\frac{\pi }{2}]$, then the possible value(s) of $y$ is/are

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

Q. The value of ${\tan }^6\frac{\pi }{9}-33{\tan }^4\frac{\pi }{9}+27{\tan }^2\frac{\pi }{9}$ is

Q.

$\frac{\sin 2A}{1+\cos 2A}$ =

1. $\tan A$

2. $\cot A$

3. $\frac{2\tan \frac{A}{2}}{1-{\tan }^2\frac{A}{2}}$

4. $\tan 2A$