Conditional Identities

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. 1

2. 1

3. = 0

4. = 1

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

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

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

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

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.

If $A+B+C={180}^{\circ }$. Find ${\sin }^{2}A+{\sin }^{2}B+{\sin }^{2}C$ =

1. $-1-4\cos A\cos B\cos C$

2. $1+2\cos A\cos B\cos C$

3. $2+2\cos A\cos B\cos C$

4. $4\sin A\sin B\sin C$

Q.

If $a{\cos }^3\alpha +3a\cos \alpha {\sin }^2\alpha =m$ and

$a{\sin }^3\alpha +3a{\cos }^2\alpha \sin \alpha =n$, Then $(m+n{)}^{\frac{2}{3}}+(m-n{)}^{\frac{2}{3}}$

is equal to

1. $2a^2$

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

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

4. $2a^3$

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.

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.

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

1. I(n+2)

2. I(n+3)

3. I(n+4)

4. I(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. -2

3. -1

4. 2