Subject Experts Solutions for Chapter: Measurement of Angles and Trigonometric Ratios, Exercise 1: Practice Exercise

What is $\frac{\cos 7x-\cos 3x}{\sin 7x-2\sin 5x+\sin 3x}$ equal to?

In a $\Delta ABC,$ if $\sin A-\cos B=\cos C$, then what is $B$ equal to?

If $\frac{\sin \left(x+y\right)}{\sin \left(x-y\right)}=\frac{a+b}{a-b},$ then what is $\frac{\tan x}{\tan y}$ equal to?

If $\sin A\sin \left(60°-A\right)\sin \left(60°+A\right)=k\sin 3A,$ then what is $k$ equal to?

The line $y=\sqrt{3}$ meets the graph $y=\tan x,$ where $x\in \left(0,\frac{\pi }{2}\right),$ in $k$ points. What is $k$ equal to?

Let $\theta$ be a positive angle. If the number of degrees in $\theta$ is divided by the number of radians in $\theta ,$ then an irrational number $\frac{180}{\mathrm{\pi }}$ results. If the number of degrees in $\theta$ is multiplied by the number of radians in $\theta ,$ then an irrational number $\frac{125\pi }{9}$ results. The angle $\theta$ must be equal to 

If $A=\left(\cos 12°-\cos 36°\right)\left(\sin 96°+\sin 24°\right)$ and $B=\left(\sin 60°-\sin 12°\right)\left(\cos 48°-\cos 72°\right),$ then what is $\frac{A}{B}$ equal to ? 

Consider the following statements

I. If $ABC$ is an equilateral triangle, then $3\tan (A+B)\tan C=1$.

II. If $ABC$ is a triangle in which $A=78°, B=66°$, then $\tan \left(\frac{A}{2}+C\right)<\tan A$.

III. If $ABC$ is any triangle, then $\tan \left(\frac{A+B}{2}\right)\sin \left(\frac{C}{2}\right)<\cos \left(\frac{C}{2}\right)$.

Which of the above statement(s) is/are correct?