Conditional Identities

Q. If ${\sin }^4\alpha +4{\cos }^4\beta +2=4\sqrt{2}\sin \alpha \cos \beta$;
$\alpha ,\beta \in [0,\pi ]$, then $\cos (\alpha +\beta )-\cos (\alpha -\beta )$ is equal to :

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

Q.

If $\tan \left(\frac{\alpha \pi }{4}\right)=cot\left(\frac{\beta \pi }{4}\right)$, then

1. $\alpha +\beta =0$

2. $\alpha +\beta =2n$

3. $\alpha +\beta =2n+1$

4. $\alpha +\beta =2\left(2n+1\right)$, for all $n$ is an integer.

Q. $\mathrm{In}\mathrm{triangle}\mathrm{ABC},\mathrm{if}\mathrm{cotA}.\mathrm{cotB}=\frac{1}{2}\mathrm{and}\mathrm{cotB}.\mathrm{cotC}=\frac{1}{18},\mathrm{then}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{tanB}\mathrm{is}$

1. 
   3
2. 
   4
3. 
   5
4. 
   6

Q.

Explain $\tan 2x$ formula.

$\begin{array}{l}Tan2x=\frac{2tanx}{1-ta{n}^{2}x}\end{array}$

Q. $\mathrm{If}A+B+C=\pi \mathrm{and}{\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=a+\mathrm{bcosA}.\mathrm{cosB}.\mathrm{cosC},\mathrm{then}ab\mathrm{is}\mathrm{equal}\mathrm{to}$
.

1. 1
2. -1
3. 2
4. -2

Q.

In $\Delta ABC,ccos(A-\alpha )+\alpha cos(C+\alpha )=$

1. $\alpha cos\alpha$

2. $bcos\alpha$

3. $ccos\alpha$

4. $2bcos\alpha$

Q.

In $\Delta ABC,\angle C=\frac{2\pi }{3}$, then the value of $co{s}^{2}A+co{s}^{2}B-cosA.cosB=$___

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

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

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

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

Q. If in triangle ABC, tan A + tan B + tan C = 6 and tan A tan B = 2, then the triangle is

1. Right angled isosceles
2. Aacute angled isosceles
3. Aacute angled
4. equilateral

Q. In a $△ABC$, if $\cos A+\cos B+\cos C=\frac{3}{2}$, then the triangle is

1. an isosceles
2. an equilateral
3. a scalene
4. a right angled

Q. In $△ABC$, if $A,B$ and $C$ represent the angles of a triangle, then the maximum value of $\sin \frac{A}{2}+\sin \frac{B}{2}+\sin \frac{C}{2}$ is

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

Q. In a $\Delta ABC$, if $A=\frac{\pi }{4}$ and $\tan B\tan C=k$
then $k$ must satisfy

1. $k^2-6k+1\ge 0$
2. $k^2-6k+1=0$
3. $k^2-6k+1\le 0$
4. $3-2\sqrt{2}<k$

Q.

In $\Delta ABC,\angle C=\frac{2\pi }{3}$, then the value of $co{s}^{2}A+co{s}^{2}B-cosA.cosB=$___

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

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

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

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

Q. ${\cos }^{-1}\left(\cos \right(-5\left)\right)+{\sin }^{-1}\left(\sin \right(6\left)\right)-{\tan }^{-1}\left(\tan \right(12\left)\right)$ is equal to
(The inverse trigonometric functions take the principal values)

1. $4\pi -11$
2. $3\pi +1$
3. $4\pi -9$
4. $3\pi -11$

Q.

If A + B + C = $\pi$ then $ta{n}^2\frac{A}{2}$ + $ta{n}^2\frac{B}{2}$ + $ta{n}^2\frac{C}{2}$ is always

1. $\le$ 1

2. $\ge$ 1

3. = 0

4. = 1

Q. In a $△ABC,\tan A=\frac{1}{2},\tan B=k+\frac{1}{2}$ and $\tan C=2k+\frac{1}{2}$, then the value of $\left[k\right]$ is
(where [.] represents greatest integer function)

Q. If $A,B,C$ are angles of a triangle and angle $A$ is obtuse, then the exhaustive set of value of $\tan B\tan C$ is

1. $(0,1)$
2. $(0,2)$
3. $(0,0.5)$
4. any positive real number

Q.

If $A+B+C=\pi$ then, $\tan \frac{A}{2}\cdot \tan \frac{B}{2}+\tan \frac{B}{2}\cdot \tan \frac{C}{2}+\tan \frac{C}{2}\cdot \tan \frac{A}{2}$ =

1. $\tan \frac{A}{2}$ . $\tan \frac{B}{2}$ . $\tan \frac{C}{2}$

2. $\tan \frac{A}{2}$ + $\tan \frac{B}{2}$ + $\tan \frac{C}{2}$

3. $-\tan \frac{A}{2}$. $\tan \frac{B}{2}$ . $\tan \frac{C}{2}$

4. $1$

Q.

If $\frac{cos(x-y)}{cos(x+y)}+\frac{cos\left(7_t\right)}{cos(7-t)}=0$ then tan x tan y tan 7 tan t=

1. 1

2. -1

3. 2

4. -2

Q. If $A,B,C$ are the three angles in a triangle such that $2\sin B\sin (A+B)-\cos A=1$ and $2\sin C\sin (B+C)-\cos B=0$, then

1. $A={120}^{\circ }$
2. $B={120}^{\circ }$
3. $C={30}^{\circ }$
4. $B=C$

Q.

Let I (n)=2cos n x, $nϵN$, then I(1)I(n+1)-I(n)=___

1. I(n+4)

2. I(n+2)

3. I(n+3)

4. I(1)

Q. If A + B + C = ${90}^{\circ }$ then $\frac{cotA+cotB+cotC}{cotAcotBcotC}$

1. 1
2. -1
3. \N
4. 2

Q. If $A,B,C$ are the angles of triangle such that $0<A\le \frac{\pi }{3}$, then the range of $\frac{\tan B+\tan C}{\tan B\tan C-1}$ is

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

Q. If $A+B+C=\pi$, then find the value of
$\frac{\cos A}{\sin B\sin C}+\frac{\cos B}{\sin A\sin C}+\frac{cosC}{sinBsinA}$ is

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

Q. If A, B, C be the angles of a triangle, then $\sum \frac{cotA+cotB}{tanA+tanB}=$

1. 1
2. 2
3. -1
4. -2

Q. In triangle $ABC,\frac{\sin A+\sin B+\sin C}{\sin A+\sin B-\sin C}$ is equal to

1. $\tan \frac{A}{2}\cot \frac{B}{2}$
2. $\cot \frac{A}{2}\tan \frac{B}{2}$
3. $\cot \frac{A}{2}\cot \frac{B}{2}$
4. $\tan \frac{A}{2}\tan \frac{B}{2}$

Q. If in $△ABC,\angle A=\frac{\pi }{4}$ and $\tan B\tan C=p$, then the possible set of value(s) of $p$ is/are

1. $(-\infty ,0)$
2. $\left(\right(\sqrt{2}+1{)}^2,\infty )$
3. $\left[\right(\sqrt{2}+1{)}^2,\infty )$
4. $(-\infty ,0]$