Trigonometric Ratios of Multiples of an Angle

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. 
2. 
3. 
4. 

Q.

If tan $\frac{A}{2}$ = $\frac{3}{2}$, then $\frac{1+cosA}{1-cosA}$ =

1. -5

2. 5

3. 9/4

4. 4/9

Q. Let f$\left(\theta \right)=sin\theta (sin\theta +sin3\theta ).Thenf\left(\theta \right)$is

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

Q.

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

${\sum }_{r=0}^{7}ta{n}^2\left(\frac{\pi r}{16}\right)=$___

1. 28

2. 35

3. 1

4. 3

Q.

Find the value of sin3θ/(1+2cos2θ)

cos

cos

sin

cos

Q.

$tan\alpha +2tan2\alpha +4tan4\alpha +8cot8\alpha$=

[IIT 1988; MP PET 1991]

1. 

2. 

3. 

4. 

Q.

If $f\left(\theta \right)=si{n}^2\theta +si{n}^2(\theta +\frac{2\pi }{3})+si{n}^2(\theta +\frac{4\pi }{3}),thenf\left(\frac{\pi }{15}\right)$

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

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

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

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

Q.

Find the value of $\frac{co{s}^{3}A-cos3A}{cosA}$ + $\frac{si{n}^{3}A-sin3A}{sinA}$

__

Q.

cosec A - 2cot 2A cos A =

1. 2sinA

2. secA

3. None of these

4. 2cosAcotA