Question

Find the value of sin $(\pi +x)sin(\pi -x)$ ${cosec}^{2}x$

Answer 1

We know that $\pi +\theta$ is in third quadrant.$\therefore sin(\pi +\theta )$ is negative in third quadrant by ASTC rule.

and $\pi -\theta$ is in second quadrant.$\therefore sin(\pi -\theta )$ is positive in second quadrant by ASTC rule.

$\sin (\pi +x)\sin (\pi -x){\csc }^{2}x$

$=-\sin x\sin x{\csc }^{2}x$

$=-{\sin }^{2}x{\csc }^{2}x$

$=-1$ by reciprocal relations of trigonometry.