If $\tan A=\frac{1}{\sqrt{3}}$, find all other trigonometric ratios of angle $A$.

Write 'True' or 'False and justify your answer.

$\sqrt{\left(1-{\cos }^2\theta \right){\sec }^2\theta }=\tan \theta$

Write 'True' or 'False and justify your answer.

$(\tan \theta +2)(2\tan \theta +1)=5\tan \theta +{\sec }^2\theta$

Write 'True' or 'False and justify your answer.

If $\cos A+{\cos }^{2}A=1$, then ${\sin }^{2}A+{\sin }^{4}A=1$.

Prove the following identity, where the angles involved are acute angles for which the trigonometric ratios as defined:

$\frac{1+{\tan }^2 \mathrm{A}}{1+{\cot }^2 \mathrm{A}}={\tan }^2 \mathrm{A}$

Prove the following identity, where the angles involved are acute angles for which the trigonometric ratios as defined:

$\frac{{\sin }^3\theta +{\cos }^3\theta }{\sin \theta +\cos \theta }=1-\sin \theta \cos \theta$

Prove the following identity, where the angles involved are acute angles for which the trigonometric ratios as defined:

$(\sec A+\tan A)(1-\sin A)=\cos A$

Prove the following identity, where the angles involved are acute angles for which the trigonometric ratios as defined:

$\left(1+{\tan }^2\mathrm{A}\right)(1-\mathrm{sinA})(1+\mathrm{sinA})=1$