Sum of Cosines of Angles in Arithmetic Progression

Q. For a positive integer $n,$ let $f_n\left(\theta \right)=\left(\tan \frac{\theta }{2}\right)(1+\sec \theta )(1+\sec 2\theta )(1+\sec 4\theta )...(1+\sec 2^n\theta ).$
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. If $r:R:r_1=2:5:12,$ then $\angle A=$

1. ${45}^{\circ }$
2. ${60}^{\circ }$
3. ${30}^{\circ }$
4. ${90}^{\circ }$

Q. Evaluate the limit:
$\underset{x\to \pi }{lim}\frac{1-\sin \frac{x}{2}}{\cos \frac{x}{2}(\cos \frac{x}{4}-\sin \frac{x}{4})}$

Q.

If n is a positive integer then $(1+i{)}^n+(1-i{)}^n$ =

1. 

2. 

3. 

4. 

Q. The value of $\sum _{n=1}^{10}(\sin \frac{2n\pi }{11}-\cos \frac{2n\pi }{11})$ is equal to

Q. The value of $(1+\sec {10}^{\text{o}})(1+\sec {20}^{\text{o}})(1+\sec {40}^{\text{o}})(1+\sec {80}^{\text{o}})$ is

1. $\tan {80}^{\text{o}}\cot 5^{\text{o}}$
2. $\tan {80}^{\text{o}}\cot {10}^{\text{o}}$
3. $\tan {40}^{\text{o}}\cot 5^{\text{o}}$
4. $\tan {40}^{\text{o}}\cot {10}^{\text{o}}$

Q. Find the value of ${\sin }^{-1}\left(\sin \frac{3\pi }{5}\right).$

Q. Find the value of the trigonometric function
$\cot (-\frac{15\pi }{4})$.

Q. If $\tan x=\frac{3}{4},\pi <x<\frac{3\pi }{2}$, find the value of $\sin \frac{x}{2},\cos \frac{x}{2}$ and $\tan \frac{x}{2}.$

Q.

If A+B+C=90degrees then sin2A+sin2B+sin2C=?

Options 1)cosAcosBcosC

2)2cosAcosBcosC

3)3cosAcosBcosC

4)4cosAcosBcosC

Answer is option 4

Q. How do you evaluate ${\sec }^{-1}2?$

Q. The number of solution(s) of ${\cos }^{5}x+{\cos }^5(x+\frac{2\pi }{3})+{\cos }^5(x+\frac{4\pi }{3})=0$ in $[0,2\pi ]$ is

1. $0$
2. $4$
3. $6$
4. $10$

Q. general solution of sin 2x +cos x=0 please expalin step by step?????

Q. If $sec\theta -tan\theta =k,$ then the value of $sec\theta +tan\theta$ is

1. $(1-k)$
2. $(1+k)$
3. $\frac{1}{k}$
4. $1-\frac{1}{k}$

Q. $\mathrm{The}\mathrm{value}\mathrm{of}\cos \left(\frac{\pi }{11}\right)+\cos \left(\frac{3\pi }{11}\right)+\cos \left(\frac{5\pi }{11}\right)+\cos \left(\frac{7\pi }{11}\right)+\cos \left(\frac{9\pi }{11}\right)\mathrm{is}$

1. 
   0
2. 
   0.5
3. 
   1
4. 
   2

Q. Show that $(1+\cos \frac{\pi }{9})(1+\cos \pi \frac{3\pi }{9})(1+\cos \frac{5\pi }{9})(1+\cos \frac{7\pi }{9})=\frac{9}{16}$.

Q. The period of $si{n}^2\theta$ is

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

Q. If $f\left(x\right)=\frac{\tan \left({\displaystyle \frac{\mathrm{\pi }}{4}}-x\right)}{\cot 2x}$ for x ≠ π/4, find the value which can be assigned to f(x) at x = π/4 so that the function f(x) becomes continuous every where in [0, π/2].

Q. Prove that
$\tan (\frac{\pi }{4}+\theta )+\tan (\frac{\pi }{4}-\theta )=2\sec 2\theta$

Q. $\underset{x\to \pi /2}{lim}\frac{\cot x-\cos x}{{\left(\pi -2x\right)}^3}$ equals

1. $\frac{1}{24}$
2. $\frac{1}{16}$
3. $\frac{1}{8}$
4. $\frac{1}{4}$

Q. For non-negative integers $n,$ let
$f\left(n\right)=\frac{\sum _{k=0}^n\sin \left(\frac{k+1}{n+2}\pi \right)\sin \left(\frac{k+2}{n+2}\pi \right)}{\sum _{k=0}^n{\sin }^2\left(\frac{k+1}{n+2}\pi \right)}$
Assuming ${\cos }^{-1}x$ takes values in $[0,\pi ],$ which of the following options is/are correct?

1. $\sin \left(7{\cos }^{-1}f\right(5\left)\right)=0$
2. $\text{If}\alpha =\tan \left({\cos }^{-1}f\right(6\left)\right),\text{then}{\alpha }^2+2\alpha -1=0$
3. $\underset{n\to \infty }{lim}f\left(n\right)=\frac{1}{2}$
4. $f\left(4\right)=\frac{\sqrt{3}}{2}$

Q.

Find the value of cos$\frac{\pi }{6}$.cos$\frac{\pi }{3}$.cos$\frac{4\pi }{6}$.cos$\frac{8\pi }{6}$.cos$\frac{16\pi }{6}$.cos$\frac{32\pi }{6}$

1. $\frac{\sqrt{3}}{32}$

2. $\frac{\sqrt{3}}{64}$

3. $\frac{\sqrt{3}}{8}$

4. $\frac{\sqrt{3}}{16}$

Q. For a positive integer $n,$ let $f_n\left(\theta \right)=\left(\tan \frac{\theta }{2}\right)(1+\sec \theta )(1+\sec 2\theta )(1+\sec 4\theta )...(1+\sec 2^n\theta ).$
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. If ${\int }_0^{1}co{t}^{-1}(1-x+x^2)dx=\lambda {\int }_0^{1}ta{n}^{-1}xdx$ then $\lambda$ is equal to

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

Q. ${\cos }^{2}x\frac{dy}{dx}+y=\tan x$

Q. $\int \frac{1-\cos x}{1+\cos x}dx$

Q. If $I_n=\underset{0}{\overset{\frac{\pi }{2}}{\int }}co{s}^{n}xcosnxdx,then\sqrt{\frac{I_4}{I_8}}$ is equal to

Q.

$se{c}^2\left[co{t}^{-1}\left(\frac{1}{2}\right)\right]+cose{c}^2\left[ta{n}^{-1}\left(\frac{1}{3}\right)\right]=$

1. 15
   

2. 20
   

3. 25

4. 10
   

Q. If ${\tan }^{-1}(1+x)+{\tan }^{-1}(1-x)=\frac{\pi }{2}$ then $x=$?

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

Q. Integrate the function $\sqrt{\frac{1-\sqrt{x}}{1+\sqrt{x}}}$