In a right triangle right angled at $\mathrm{B}$, if $4 \mathrm{cosec} \mathrm{A}=5$ then verify that $1+{\tan }^2 \mathrm{A}={\sec }^2 \mathrm{A}$.

If $4 \tan \theta =3$, evaluate $\frac{4 \sin \theta -\cos \theta +1}{4 \sin \theta +\cos \theta -1}$.

If $\angle \mathrm{A}$ and $\angle \mathrm{P}$ are acute angles such that $\cos \mathrm{A}=\cos \mathrm{P}$, then show that $\angle \mathrm{A}=\angle \mathrm{P}$.

In the adjoining figure, find the value of $\sec \theta$.

From the adjoining figure, find the value of $\sin \alpha +\sin \theta$.

From the adjoining figure, find the value of ${\sec }^2 \mathrm{\theta }-{\tan }^2 \mathrm{\theta }$.

From the adjoining figure, find the value of ${\cot }^2 \mathrm{\alpha }-{\mathrm{cosec}}^2 \mathrm{\alpha }$.

In $△\mathrm{ABC}, \angle \mathrm{B}=90°$. If $\mathrm{BC}=5 \mathrm{cm}$ and $\mathrm{AC}-\mathrm{AB}=1 \mathrm{cm}$, evaluate $\frac{1+\sin \mathrm{C}}{\cos \mathrm{C}}$.

If $\tan \theta =\frac{a}{b}$, prove that $\frac{a \sin \theta -b \cos \theta }{a \sin \theta +b \cos \theta }=\frac{a^2-b^2}{a^2+b^2}$.