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

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\frac{\cos A-\sin A+1}{\cos A+\sin A-1}=\operatorname{cosec} A+\cot A,$ using the identity $\operatorname{cosec}^{2} A=1+\cot ^{2} A$

In a right triangle $A B C$, right-angled at $B$. if $\tan A =1,$ then verify that $2 \sin A \cos A=1$

In $Fig.$ find $\tan P-\cot R .$

In triangle $ABC ,$ right -angled at $B ,$ if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:

$(i)$ $\sin A \cos C+\cos A \sin C$

$(ii)$ $\cos A \cos C-\sin A \sin C$

Evaluate:

$\frac{\sin 18^{\circ}}{\cos 72^{\circ}}$