Conditional Identities

If in a $∆ABC, \tan A+\tan B+\tan C>0,$ then the triangle is

If $A+B+C=\pi ,n\in Z,$ then $\tan nA+\tan nB+\tan nC$ is equal to

In any triangle $ABC$, which is not right angled $\Sigma \text{ cos }\text{A }\text{. }\text{cosec }\text{B }\text{. }\text{cosec }\text{C}$  is equal to

If $\alpha , \beta , \gamma$ are real numbers satisfying $\alpha +\beta +\gamma =\pi ,$ the minimum value of the given expression $\sin \alpha +\sin \beta +\sin \gamma$ is

If $A+B+C=\mathrm{\pi }$, then prove that $\tan A+\tan B+\tan C=\tan A.\tan B.\tan C$.

If $A, B, C$ are angles of $∆ABC$ and  $\tan A\tan C=3, \tan B\tan C=6,$ then 

If $ABC$ is a triangle and $\tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2}$ are in H.P., then the minimum value of $\cot \frac{B}{2}$ is equal to

If ${\sin }^{-1}a+{\sin }^{-1}b+{\sin }^{-1}c=\pi$ , then the value of $a \cdot \sqrt{\left(1-a^2\right)}+b\cdot \sqrt{\left(1-b^2\right)}+c\cdot \sqrt{\left(1-c^2\right)}$ will be

In a triangle ABC, $\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{A}\mathrm{ }+\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{B}\mathrm{ }+\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{C}\mathrm{ }=\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{\theta }$ .

$\frac{\sin \mathrm{A}\sin \mathrm{B}\sin \mathrm{C}}{1+\cos \mathrm{A}\cos \mathrm{B}\cos \mathrm{C}}=$

In a triangle $\mathrm{ABC}$, if $\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{A}\mathrm{ }+\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{B}\mathrm{ }+\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{C}\mathrm{ }=\mathrm{ }\mathrm{c}\mathrm{o}\mathrm{t}\mathrm{\theta }$, then the possible value of $\mathrm{\theta }$  is-

If $x+y+z=xyz$ , then the value of $x\left(1-y^2\right) \left(1-z^2\right)+y\left(1-x^2\right)\left(1-z^2\right)+z\left(1-x^2\right)\left(1-y^2\right)$ is

Statement - 1: The minimum value of the expression $\sin \alpha +\sin \beta +\sin \gamma$ where $\alpha , \beta , \gamma$ are real numbers such that $\alpha +\beta +\gamma =\pi$ , is not – 3 

Statement - 2: $\alpha , \beta , \gamma$ are angles of a triangle

In a $\mathrm{\Delta }ABC,$ if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A.\sin B.\sin C=\frac{3+ \sqrt{3}}{8}$ , then on the basis of above information, answer the following questions:

The value of $\tan A+\tan B+\tan C$ is:

The angles of $\mathrm{\Delta }ABC$ are :

In a $\mathrm{\Delta }ABC,$ if $\cos A\cos B\cos C=\frac{\sqrt{3}-1}{8}$ and $\sin A.\sin B.\sin C=\frac{3+ \sqrt{3}}{8}$ , then on the basis of above information, answer the following questions:

The value of $\tan A+\tan B+\tan C$ is:

In a triangle $ABC$, if $a = 7, b = 8, c = 9$, then the length of the line joining $B$ to the midpoint of $AC$ is

If $\tan A:\tan B:\tan C=1:2:3$, $A+B+C=\mathrm{\pi }$, then $\sin A:\sin B:\sin C$ is equal to

The angular elevation of a tower $CD$ at a point $A$ due south of it is $60°$, and at a point $B$ due west of $A$, the elevation is $30°$. If $AB=3 km$, the height of the tower is:

If in $\mathrm{\Delta }ABC$, $\tan A+\tan B+\tan C=x^2+6x+5$, then set of values of $x$ for which $\mathrm{\Delta }ABC$ is obtuse angled,

If $A+B+C=\pi$ and $\sum \cos A\mathrm{cosec}B\mathrm{cosec}C=k,$ then $k$ is equal to