Properties Derived from Trigonometric Identities

Q.

If $\sin A=\frac{1}{\sqrt{10}}$ and $\sin B=\frac{1}{\sqrt{5}}$, where $A$ and $B$ are positive acute angles, then $A+B$ is equal:

1. $\mathrm{\pi }$

2. $\frac{\mathrm{\pi }}{2}$

3. $\frac{\mathrm{\pi }}{3}$

4. $\frac{\mathrm{\pi }}{4}$

Q. The range of the function
$f\left(x\right)={\log }_{\sqrt{5}}(3+\cos (\frac{3\pi }{4}+x)+\cos (\frac{\pi }{4}+x)+\cos (\frac{\pi }{4}-x)-\cos (\frac{3\pi }{4}-x))$
is

1. $[0,2]$
2. $[-2,2]$
3. $(0,\sqrt{5})$
4. $[\frac{1}{\sqrt{5}},\sqrt{5}]$

Q.

Is $\cot$ the same as $\tan$ inverse of $1?$

Q. Let $M$ and $m$ respectively be the maximum and minimum values of the function $f\left(x\right)={\tan }^{-1}(\sin x+\cos x)\text{in}[0,\frac{\pi }{2}].$ Then the value of $\tan (M-m)$ is equal to

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

Q.

$2\pi -\left({\sin }^{-1}\left(\frac{4}{5}\right)+{\sin }^{-1}\left(\frac{5}{13}\right)+{\sin }^{-1}\left(\frac{16}{65}\right)\right)$is equal to

1. $\frac{5\pi }{4}$

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

3. $\frac{7\pi }{4}$

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

Q.

The value of ${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)-{\sin }^{-1}\left(\frac{1}{2}\right)$ is

1. $45°$

2. $90°$

3. $15°$

4. $30°$

Q. If ${\tan }^{-1}(x+1)+{\tan }^{-1}(x-1)={\tan }^{-1}\left(\frac{8}{31}\right)$, then $x$ is equal to

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

Q.

Formula of $cot\left(3x\right)$ is

Q.

If $y={\sin }^{-1}\frac{2x}{1+x^2}+se{c}^{-1}\frac{1+x^2}{1-x^2}$, then $\frac{dy}{dx}=$

1. $\frac{4}{\left(1-x^2\right)}$

2. $\frac{4}{\left(1+x^2\right)}$

3. $\frac{1}{\left(1+x^2\right)}$

4. $\frac{-4}{\left(1+x^2\right)}$

Q. The value of ${\cos }^2{45}^{\circ }-{\sin }^2{15}^{\circ }$ is:

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

Q.

How do you find the exact value of ${\tan }^{-1}\left(0\right)$?

Q. The fundamental period of the function $y=\sin x$ is

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

Q. Write the following functions in the simplest form :
${\tan }^{-1}\left(\frac{1}{\sqrt{x^2-1}}\right),|x|>1$

Q.

Show that $tan\left(\frac{1}{2}si{n}^{-1}\frac{3}{4}\right)=\frac{4-\sqrt{7}}{3}$ and justify why the other value $\frac{4+\sqrt{7}}{3}$ is ignored?

Q. Write the following functions in the simplest form: ${\tan }^{-1}\frac{\sqrt{1+x^2}-1}{x},x\ne 0$

Q.

The function $f\left(x\right)=x+\sin x$ has

1. A minimum but no maximum

2. A maximum but no minimum

3. Neither maximum nor minimum

4. Both maximum and minimum

Q.

The maximum value of $(se{c}^{-1}x{)}^2+(cose{c}^{-1}x{)}^2$ is equal to

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

2. $\frac{5{\pi }^2}{4}$

3. ${\pi }^2$

4. None of these

Q.

The sides of a triangle are 3x+4y, 4x+3y, 5x+5y where x, y>0 then Triangle is

1.Right angle

2.Obtuse angle

3.Equilateral

4.None of the above

Q. The period of the function $y={\sin }^{-1}\left(\sin x\right)$ is

1. $2\pi$
2. $4\pi$
3. $3\pi$
4. $1\pi$

Q. The sum of the infinite terms of the series ${\cot }^{-1}(1^2+\frac{3}{4})+{\cot }^{-1}(2^2+\frac{3}{4})+{\cot }^{-1}(3^2+\frac{3}{4})+\cdots$ is equal to

1. ${\tan }^{-1}1$
2. $\pi -{\tan }^{-1}2$
3. ${\tan }^{-1}2$
4. $-{\tan }^{-1}1$

Q.

Let $\alpha ,\beta$be such that $\pi <(\alpha -\beta )<3\pi$. If $\sin \alpha +\sin \beta =\frac{-21}{65}$ and $cos\alpha +\cos \beta =\frac{-27}{65}$, then the value of$\frac{\cos (\alpha -\beta )}{2}$ is:

1. $\frac{-6}{65}$

2. $\frac{3}{\sqrt{130}}$

3. $\frac{6}{65}$

4. $-\frac{3}{\sqrt{130}}$

Q. Show that

$2{\sin }^{-1}\left(\frac{3}{5}\right)-{\tan }^{-1}\left(\frac{17}{31}\right)=\frac{\pi }{4}$

Q.

The derivative of ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$ with respect to ${\cos }^{-1}\sqrt{1-x^2}$ is

1. $\sqrt{\frac{1-x^2}{1+x^2}}$

2. $\frac{1}{\sqrt{1-x^2}}$

3. $\frac{2}{1+x^2}$

4. $\frac{2\sqrt{1-x^2}}{1+x^2}$$={\cos }^{-1}\sqrt{{\cos }^{2}x}$

Q.

If $y=co{t}^{-1}\tan \left(\begin{array}{l}\frac{x}{2}\end{array}\right)$, then $\frac{dy}{dx}$ is equal to?

1. $0$

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

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

4. $\frac{x}{2}$

Q.

If ${\tan }^{-1}2x+{\tan }^{-1}3x=\frac{\pi }{4}$ then $x=$

1. $-1$

2. $\frac{1}{6}$

3. $-1,\frac{1}{6}$

4. None of these

Q.

If $x$ takes a negative permissible value, then ${\sin }^{-1}x$ is equal to :

1. $–{\cos }^{-1}\sqrt{1-x^2}$

2. ${\cos }^{-1}\sqrt{x^2-1}$

3. $\mathrm{\pi }–{\cos }^{-1}\sqrt{1-x^2}$

4. ${\cos }^{-1}\sqrt{1-x^2}$

Q.

If $co{s}^{-1}p+co{s}^{-1}q+co{s}^{-1}r=\pi$ then $p^2+q^2+r^2=$

1. 1
   

2. pqr
   

3. 1+pqr
   

4. 1-2pqr

Q.

The value of $sin\left[ta{n}^{-1}\right(-\sqrt{3})+co{s}^{-1}(-\frac{\sqrt{3}}{2})]$

1. 1

2. -1

3. 0

4. None of these

Q. If $\lambda$ is a non real cube root of -2, then the value of $\left|\begin{array}{ccc}1& 2\lambda & 1\\ \lambda & 1& 3{\lambda }^2\\ 2& 2\lambda & 1\end{array}\right|$
is equal to

1. -11
2. -12
3. -13
4. 0

Q.

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

1. 5
2. 13
3. 15
4. 6