Conditional Identities

If the line $x=y=z$ intersects the line $x \sin A+y \sin B+z \sin C-18=0=x \sin 2A+y \sin 2B+z\sin 2C-9$, where $A, B, C$ are the angles of a triangle $ABC$, then $80\left(\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}\right)$ is equal to _________.

The value of $\left(\frac{\sin 148°+\sin 56°+\sin 156°}{\left(\cos 74°+\cos 28°+\cos 78°-1\right)\left(\cos 37°\cos 14°\cos 39°\right)}\right)$ is

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:

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

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

If in a$∆ABC, \angle C=90°,$then the maximum value of $\sin A\sin B$is

In a $∆ABC,\cos \left(\frac{B+2C+3A}{2}\right)+\cos \left(\frac{A-B}{2}\right)$ is equal to

If $A+B+C=\pi$, then ${\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C$$+2\cos A\cos B\cos C$ is equal to

Find value of $\mathrm{\Sigma cosA}.\mathrm{cosecB}.\mathrm{cosecC}$ ,for any triangle $\mathrm{ABC}$ which is not right angled.

There exist a $∆ABC$ satisfying

If the line $x=y=z$ intersects the line $\sin A·x+\sin B·y+\sin C·z=2d^2$, $\sin 2A·x+\sin 2B·y+\sin 2C·z=d^2$ then $\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$ is equal to (where $A+B+C=\pi$)