Find an acute angle $\mathrm{\theta }$ when, $\frac{\cos \theta -\sin \theta }{\cos \theta +\sin \theta }=\frac{1-\sqrt{3}}{1+\sqrt{3}}$.

If, $\tan \theta =cot(30°+\theta ),$ then find the value of $\mathrm{\theta }.$

Prove that $(\mathrm{tan\theta }+2)(2\mathrm{tan\theta }+1)=5\mathrm{tan\theta }+2{\sec }^2\mathrm{\theta }$.

Prove that:

$\frac{\cos \mathrm{A}}{1+\sin \mathrm{A}}+\tan \mathrm{A} = \sec \mathrm{A}$

Prove that:

$\frac{\sin \mathrm{\theta }}{1-\mathrm{cot\theta }}+\frac{\cos {\displaystyle \mathrm{\theta }}}{1-\mathrm{tan\theta }}=\mathrm{cos\theta }+\mathrm{sin\theta }$

Prove that:

$\frac{{\tan }^2\theta }{(\mathrm{sec\theta }-1{)}^2}=\frac{1+\cos \theta }{1-\cos \theta }$

If $\mathrm{cot\theta } = \frac{15}{8}$,then find the value of

$\frac{(2+2\sin \theta )(1-\sin \theta )}{(1+\cos \theta )(2-2\cos \theta )}$

If $4\mathrm{x}=\mathrm{cosec\theta }$ and $\frac{4}{\mathrm{x}}=\mathrm{cot\theta }$, then find the value of

$4({\mathrm{x}}^2-\frac{1}{{\mathrm{x}}^2})$