Exercise

If $\sin \theta =\mathrm{cosec}\theta$ and $0\le \theta \le \pi$, the value of $\theta$ is

A kite string is $48 \mathrm{m}$ long. As the wind blew, the angle between the kite and the ground went from $27$ degrees to $54$ degrees. Determine the increase in the vertical height of the kite above the ground.

If $\frac{{\sin }^4\alpha }{a}+\frac{{\cos }^4\alpha }{b}=\frac{1}{a+b}$, then prove that $\frac{{\sin }^6\alpha }{a^2}+\frac{{\cos }^6\alpha }{b^2}=\frac{1}{(a+b{)}^2}.$.

If $4 \mathrm{tan\theta }=3$, evaluate $\left(\frac{4\mathrm{sin\theta }-\mathrm{cos\theta }+1}{4\mathrm{sin\theta }+\mathrm{cos\theta }-1}\right)$.

Prove that: $\frac{\sin A-2{\sin }^{3}A}{2{\cos }^{3}A-\cos A}=\tan A$.

Prove that $\left(1+\cot A-\mathrm{cosec}A\right)\left(1+\tan A+\sec A\right)=2$.

If $\sin \theta =\frac{a}{b}$, then $\cos \theta$ is equal to 

$\left(\sec A+\tan A\right)\left(1-\sin A\right)$ is equal to