Sin2A and Cos2A in Terms of tanA

Q.

If $\tan \left(\theta \right)-cot\left(\theta \right)=a$ and $\sin \left(\theta \right)+\cos \left(\theta \right)=b$, then ${\left(b^2-1\right)}^2\left(a^2+4\right)$ is equal to

1. $2$

2. $\pm 4$

3. $-4$

4. $4$

Q.

Prove that cot $(\frac{\pi }{4}-2co{t}^{-1}3)=7$

Q.

The value of $\cos \left(2{\tan }^{-1}\left(-7\right)\right)$ is

1. $\frac{49}{50}$

2. $-\frac{49}{50}$

3. $\frac{24}{25}$

4. $-\frac{24}{25}$

5. $\frac{48}{49}$

Q.

Prove that:

$cos4A=1-8co{s}^{2}A+8co{s}^{4}A$

Q. Let $\alpha$ and $\beta$ be nonzero real numbers such that $2(\cos \beta -\cos \alpha )+\cos \alpha \cos \beta =1.$ Then which of the following is/are true?

1. $\tan \left(\frac{\alpha }{2}\right)+\sqrt{3}\tan \left(\frac{\beta }{2}\right)=0$
2. $\sqrt{3}\tan \left(\frac{\alpha }{2}\right)+\tan \left(\frac{\beta }{2}\right)=0$
3. $\tan \left(\frac{\alpha }{2}\right)-\sqrt{3}\tan \left(\frac{\beta }{2}\right)=0$
4. $\sqrt{3}\tan \left(\frac{\alpha }{2}\right)-\tan \left(\frac{\beta }{2}\right)=0$

Q. If $\cos \theta -\sin \theta =\frac{1}{5}$, where $0<\theta <\frac{\pi }{2}$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(1)}& \frac{(\cos \theta +\sin \theta )}{2}& \left(p\right)& \frac{4}{5}\\ \text{(2)}& \sin 2\theta & \left(q\right)& \frac{7}{10}\\ \text{(3)}& \cos 2\theta & \left(r\right)& \frac{24}{25}\\ \text{(4)}& \cos \theta & \left(s\right)& \frac{7}{25}\end{array}$

Which of the following is the correct combination?

1. $1\to s,2\to p,3\to r,4\to q$
2. $1\to p,2\to s,3\to r,4\to p$
3. $1\to q,2\to s,3\to r,4\to p$
4. $1\to q,2\to r,3\to s,4\to p$

Q.

If $\alpha and\beta$ are acute angles satisfying $cos2\alpha =\frac{3cos2\beta -1}{3-cos2\beta },$ then $tan\alpha$ =

1. $\sqrt{2}tan\beta$

2. $\frac{1}{\sqrt{2}}tan\beta$

3. $\sqrt{2}cot\beta$

4. $\frac{1}{\sqrt{2}}cot\beta$

Q. For any angle $\theta ,$ In a $\Delta ABC,b\cos (C+\theta )+c\cos (B-\theta )$ is equal to

1. $a\sin \theta$
2. $a\cos \theta$
3. $a\tan \theta$
4. $a\cot \theta$

Q. Let $x,y$ be real numbers such that $x{\cos }^2{39}^{\circ }=\tan {26}^{\circ }\cot {39}^{\circ }\tan {86}^{\circ }\cos {78}^{\circ }\tan {34}^{\circ }$ and $y\cos {81}^{\circ }=\cos {36}^{\circ }-\sin {36}^{\circ }.$ Then the value of $x+y^2$ is

1. $4$
2. $5$
3. $10$
4. $15$

Q. The value of ${\cot }^2\frac{\pi }{12}-{\tan }^2\frac{\pi }{12}$ is

1. $4\sqrt{3}$
2. $8\sqrt{3}$
3. $2\sqrt{3}$
4. $\frac{8}{\sqrt{3}}$

Q. Let $\alpha$ and $\beta$ be nonzero real numbers such that $2(\cos \beta -\cos \alpha )+\cos \alpha \cos \beta =1.$ Then which of the following is/are true?

1. $\tan \left(\frac{\alpha }{2}\right)+\sqrt{3}\tan \left(\frac{\beta }{2}\right)=0$
2. $\sqrt{3}\tan \left(\frac{\alpha }{2}\right)+\tan \left(\frac{\beta }{2}\right)=0$
3. $\tan \left(\frac{\alpha }{2}\right)-\sqrt{3}\tan \left(\frac{\beta }{2}\right)=0$
4. $\sqrt{3}\tan \left(\frac{\alpha }{2}\right)-\tan \left(\frac{\beta }{2}\right)=0$

Q. There exists a positive real number $x$ satisfying $\cos \left({\tan }^{-1}x\right)=x.$ Then the value of ${\cos }^{–1}\left(\frac{x^2}{2}\right)$ is

1. $\frac{2\pi }{5}$
2. $\frac{4\pi }{5}$
3. $\frac{\pi }{10}$
4. $\frac{\pi }{5}$

Q. If $f\left(x\right)=1+\cos x+{\cos }^{2}x+\cdots \infty$, then $f\left(x\right)+f(\frac{\pi }{2}+x)+f(\frac{\pi }{2}-x)+f(\pi -x)$ equals

1. ${\sin }^{2}2x{\cos }^{2}2x$
2. $2{\sec }^{2}x+{\cos }^{2}x$
3. $2{\sec }^{2}x{\csc }^{2}x$
4. $2({\sin }^{2}x+{\csc }^{2}x)$

Q. If $tanx=\frac{2b}{a-c}(a\ne c),y=aco{s}^{2}x+2bsinxcosx+csi{n}^{2}xandz=asi{n}^{2}x-2bsinxcosx+co{s}^{2}x,$ then

1. 
2. 
3. 
4. 

Q.

Solution of equation $tan\left(co{s}^{-1}x\right)=sin\left[co{t}^{-1}\frac{1}{2}\right]$ is

1. 
   

2. 
   

3. None of these
4. 

Q. The value of ${\sin }^{-1}\left(\sin \left(\frac{17\pi }{4}\right)\right)+{\cos }^{-1}\left(\cos \left(\frac{27\pi }{5}\right)\right)+{\tan }^{-1}\left(\tan \left(\frac{37\pi }{6}\right)\right)$ is

1. $\frac{50\pi }{51}$
2. $\frac{61\pi }{60}$
3. $\frac{71\pi }{70}$
4. $\frac{41\pi }{40}$

Q. $\frac{\cos 2B-\cos 2A}{\sin 2A+\sin 2B}=\tan (A-B)$

1. True
2. False

Q. The value of ${\cot }^4\frac{\pi }{16}-4{\cot }^3\frac{\pi }{16}-6{\cot }^2\frac{\pi }{16}+4\cot \frac{\pi }{16}+2$ is

Q. For any acute angle $\theta ,\cot (\frac{3\pi }{2}+\theta )=$, $\tan (\frac{3\pi }{2}-\theta )=$

1. $-\tan \theta$
2. $\cot \theta$
3. $\tan \theta$
4. $-\cot \theta$

Q.

If cos 2 B = $\frac{cos(A+C)}{cos(A-C)}$, then tan A, tan B, tan C are in

1. A.P.

2. G.P.

3. H.P.

4. None of these

Q. The value of $\cos \left[{\tan }^{-1}\right(\tan 4\left)\right]$ is

1. $\frac{1}{\sqrt{17}}$
2. $-\frac{1}{\sqrt{17}}$
3. $\cos 4$
4. $-\cos 4$

Q. Prove that: ${\cos }^{-1}\frac{4}{5}+{\sin }^{-1}\frac{5}{13}={\cos }^{-1}\frac{33}{65}$.

Q.

Prove that:

$co{s}^2(\frac{\pi }{4}-\theta )-si{n}^2(\frac{\pi }{4}-\theta )=sin2\theta$

Q. If $\tan \alpha =2\sqrt{2},$ then the value of $\frac{\tan \alpha }{\frac{{\sin }^3\alpha }{\cos \alpha }+\sin \alpha .\cos \alpha }$ is

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

Q. Prove that
$\tan (\pi /4+\theta )-\tan (\pi /4-\theta )=2\tan 2\theta$

Q.

If $2tan\frac{\alpha }{2}=tan\frac{\beta }{2}$, prove that $cos\alpha =\frac{3+5cos\beta }{5+3cos\beta }$.

Q.

If $\frac{cosx}{a}=\frac{sinx}{b}$ then $|acos2x+bsin2x|=$

1. $|a\sqrt{ab}|$

2. $\frac{a^2}{|b|}$

3. $\frac{b^2}{|a|}$

4. |a|

Q.

If cos 2 B = $\frac{cos(A+C)}{cos(A-C)}$, then tan A, tan B, tan C are in

1. A.P.

2. H.P.

3. G.P.

4. None of these

Q.

Prove that:

$\frac{cos\theta }{1-sin\theta }=tan(\frac{\pi }{4}+\frac{\theta }{2})$

Q. If $3si{n}^{-1}\left(\frac{2x}{1+x^2}\right)-4co{s}^{-1}\left(\frac{1-x^2}{1+x^2}\right)+2ta{n}^{-1}\left(\frac{2x}{1-x^2}\right)=\frac{\pi }{3}$, where $|x|<1$, then $x$ is equal to

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