Prove that $\frac{\tan A+\sec A-1}{\tan A-\sec A+1}=\frac{1+\sin A}{\cos A}$.

If $\frac{2 \sin \alpha}{1+\cos \alpha+\sin \alpha}=y$, then prove that $\frac{1-\cos \alpha+\sin \alpha}{1+\sin \alpha}$ is also equal to $y$.

If $m \sin \theta=n \sin (\theta+2 \alpha)$, then prove that $\tan (\theta+\alpha) \cot \alpha=\frac{m+n}{m-n}$.

If $\cos (\alpha+\beta)=\frac{4}{5}$ and $\sin (\alpha-\beta)=\frac{5}{13}$, where $\alpha$ lie between 0 and $\frac{\pi}{4}$, then find that value of $\tan 2 \alpha$.

If $\tan x=\frac{b}{a}$, then find the value of $\sqrt{\frac{a+b}{a-b}}+\sqrt{\frac{a-b}{a+b}}$.

Prove that $\cos \theta \cos \frac{\theta}{2}-\cos 3 \theta \cos \frac{9 \theta}{2}=\sin 7 \theta \sin 8 \theta$

If $a \cos \theta+b \sin \theta=m$ and $a \sin \theta-b \cos \theta=n$, then show that $a^2+b^2=m^2+n^2$

Find the value of $\tan 22\Upsilon30^{\prime}$.

Prove that $\sin 4 A=4 \sin A \cos ^3 A-4 \cos A \sin ^3 A$.

If $\tan \theta+\sin \theta=m$ and $\tan \theta-\sin \theta=n, \quad$ then prove that $m^2-n^2=4 \sin \theta \tan \theta$

If $\tan (A+B)=p$ and $\tan (A-B)=q$, then show that $\tan 2 A=\frac{p+q}{1-p q}$.

If $\cos \alpha+\cos \beta=0=\sin \alpha+\sin \beta$, then prove that $\cos 2 \alpha+\cos 2 \beta=-2 \cos (\alpha+\beta)$

If $\frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}$, then show that $\frac{\tan x}{\tan y}=\frac{a}{b}$.

If $$\tan \theta = {{\sin \alpha - \cos \alpha } \over {\sin \alpha + \cos \alpha }}$$, then show that $$\sin \alpha + \cos \alpha = \sqrt 2 \cos \theta $$.

If $\sin \theta+\cos \theta=1$, then find the general value of $\theta$.

Find the most general value of $\theta$ satisfying the equation $\tan \theta=-1$ and $\cos \theta=\frac{1}{\sqrt{2}}$.

If $\cot \theta+\tan \theta=2 \operatorname{cosec} \theta$, then find the general value of $\theta$.

If $2 \sin ^2 \theta=3 \cos \theta$, where $0 \leq \theta \leq 2 \pi$, then find the value of $\theta$.

If sec $x \cos 5 x+1=0$, where $0< x \leq \frac{\pi}{2}$, then find the value of $x$.

If $\sin (\theta+\alpha)=a \quad$ and $\quad \sin (\theta+\beta)=b, \quad$ then prove that $\cos (\alpha+\beta)-4 a b \cos (\alpha-\beta)=1-2 a^2-2 b^2$.

If $\cos (\theta+\phi)=m \cos (\theta-\phi)$, then prove that $\tan \theta=\frac{1-m}{1+m} \cot \phi$

Find the value of the expression

$$3\left[\sin ^4\left(\frac{3 \pi}{2}-\alpha\right)+\sin ^4(3 \pi+\alpha)\right]-2\left[\sin ^6\left(\frac{\pi}{2}+\alpha\right)+\sin ^6(5 \pi-\alpha)\right]$$

If $a \cos 2 \theta+b \sin 2 \theta=c$ has $\alpha$ and $\beta$ as its roots, then prove that $\tan \alpha+\tan \beta=\frac{2 b}{a+c}.$

If $x=\sec \phi-\tan \phi$ and $y=\operatorname{cosec} \phi+\cot \phi$, then show that $x y+x-y+1=0$

If $\theta$ lies in the first quadrant and $\cos \theta=\frac{8}{17}$, then find the value of $\cos (30 \Upsilon+\theta)+\cos (45 \Upsilon-\theta)+\cos (120 \Upsilon-\theta)$

Find the value of $\cos ^4 \frac{\pi}{8}+\cos ^4 \frac{3 \pi}{8}+\cos ^4 \frac{5 \pi}{8}+\cos ^4 \frac{7 \pi}{8}$.

Find the general solution of the equation $5\cos^2\theta+7\sin^2\theta-6=0.$

Find the general of the equation $\sin x-3 \sin 2 x+\sin 3 x$ $=\cos x-3 \cos 2 x+\cos 3 x$.

$$\begin{aligned} &\text { Find the general solution of the equation }\\ &(\sqrt{3}-1) \cos \theta+(\sqrt{3}+1) \sin \theta=2 \end{aligned}$$

In the following match each item given under the Column I to its correct answer given under the Column II.

If $\sin \theta+\operatorname{cosec} \theta=2$, then $\sin ^2 \theta+\operatorname{cosec}^2 \theta$ is equal to

If $f(x)=\cos ^2 x+\sec ^2 x$, then

If $\tan \theta=\frac{1}{2}$ and $\tan \phi=\frac{1}{3}$, then the value of $\theta+\phi$ is

Which of the following is not correct?

The value of $$\tan 1\Upsilon\tan 2\Upsilon\tan 3\Upsilon...~\tan89\Upsilon$$ is

The value of $\frac{1-\tan ^2 15 \Upsilon}{1+\tan ^2 15 \Upsilon}$ is

The value of $\cos1\Upsilon\cos2\Upsilon\cos3\Upsilon ...~\cos179\Upsilon$ is

If $\tan \theta=3$ and $\theta$ lies in third quadrant, then the value of $\sin \theta$ is

The value of $\tan 75 \Upsilon-\cot 75\Upsilon$ is

Which of the following is correct?

If $\tan \alpha=\frac{m}{m+1}$ and $\tan \beta=\frac{1}{2 m+1}$, then $\alpha+\beta$ is equal to

The minimum value of $$3\cos x+4\sin x+8$$ is

The value of $\tan 3 A-\tan 2 A-\tan A$ is

The value of $\sin (45 \Upsilon+\theta)-\cos (45 \Upsilon-\theta)$ is

The value of $\cot \left(\frac{\pi}{4}+\theta\right) \cot \left(\frac{\pi}{4}-\theta\right)$ is

$\cos 2 \theta \cos 2 \phi+\sin ^2(\theta-\phi)-\sin ^2(\theta+\phi)$ is equal to

The value of $\cos12\Upsilon+\cos84\Upsilon+\cos156\Upsilon+\cos132\Upsilon$ is

If $\tan A=\frac{1}{2}$ and $\tan B=\frac{1}{3}$, then $\tan(2A+B)$ is equal to

The value of $\sin \frac{\pi}{10} \sin \frac{13 \pi}{10}$ is

The value of $\sin50\Upsilon-\sin70\Upsilon+\sin10\Upsilon$ is

If $\sin\theta+\cos\theta=1$, then the value of $\sin2\theta$ is

If $\alpha+\beta=\frac{\pi}{4}$, then the value of $(1+\tan \alpha)(1+\tan \beta)$ is

If $\sin \theta=\frac{-4}{5}$ and $\theta$ lies in third quadrant, then the value of $\cos \frac{\theta}{2}$ is

The number of solutions of equation $\tan x+\sec x=2 \cos x$ lying in the interval $[0,2 \pi]$ is

The value of $\sin \frac{\pi}{18}+\sin \frac{\pi}{9}+\sin \frac{2 \pi}{9}+\sin \frac{5 \pi}{18}$ is

If $A$ lies in the second quadrant and $3 \tan A+4=0$, then the value of $2 \cot A-5 \cos A+\sin A$ is

The value of $\cos ^2 48 \Upsilon-\sin ^2 12 \Upsilon$ is

If $\tan \alpha=\frac{1}{7}$ and $\tan \beta=\frac{1}{3}$, then $\cos 2 \alpha$ is equal to

$$ \text { If } \tan \theta=\frac{a}{b} \text {, then } b \cos 2 \theta+a \sin 2 \theta \text { is equal to } $$

If for real values of $x, \cos \theta=x+\frac{1}{x}$, then

60 The value of $\frac{\sin 50 \Upsilon}{\sin 130 \Upsilon}$ is .............. .

If $k=\sin \left(\frac{\pi}{18}\right) \sin \left(\frac{5 \pi}{18}\right) \sin \left(\frac{7 \pi}{18}\right)$, then the numerical value of $k$ is .................. .

$$\text { If } \tan A=\frac{1-\cos \theta}{\sin B} \text {, then } \tan 2 A= ........... .$$

If $\sin x+\cos x=a$, then

(i) $\sin ^6 x+\cos ^6 x=$ ........... .

(ii) $|\sin x-\cos x|=$ ............ .

In right angled $\Delta ABC$ with $\angle C=90\Upsilon$ the equation whose roots are $\tan A$ and $\tan B$ is ........... .

$3(\sin x-\cos x)^4+6(\sin x+\cos x)^2+4\left(\sin ^6 x+\cos ^6 x\right)=$ ............ .

Given $x>0$, the value of $f(x)=-3 \cos \sqrt{3+x+x^2}$ lie in the interval .......... .

The maximum distance of a point on the graph of the function $y=\sqrt{3} \sin x+\cos x$ from $X$-axis is ......... .

In each of the questions 68 to 75 , state whether the statements is True or False? Also, give justification.

The equality $\sin A+\sin 2 A+\sin 3 A=3$ holds for some real value of $A$.

$\sin10\Upsilon$ is greater than $\cos10\Upsilon$.

$\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{16 \pi}{15}=\frac{1}{16}$

One value of $\theta$ which satisfies the equation $\sin^4\theta-2\sin^2\theta-1$ lies between $\theta$ and $2\pi$.

If $\operatorname{cosec} x=1+\cot x$, then $x=2 n \pi, 2 n \pi+\frac{\pi}{2}$

If $\tan \theta+\tan 2 \theta+\sqrt{3} \tan \theta \tan 2 \theta=\sqrt{3}$, then $\theta=\frac{n \pi}{3}+\frac{\pi}{9}$.

If $\tan (\pi \cos \theta)=\cot (\pi \sin \theta)$, then $\cos \left(\theta-\frac{\pi}{4}\right)= \pm \frac{1}{2 \sqrt{2}}$.