Express $\cot 85^{\circ}+\cos 75^{\circ}$ in terms of trigonometric ratios of angles between $0^{\circ}$ and $45^{\circ}$

$\sin 2 A=2 \sin A$ is true when $A=$

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

$(\sin A+\operatorname{cosec} A)^{2}+(\cos A+\sec A)^{2}=7+\tan ^{2} A+\cot ^{2} A$

State whether the following are true or false. Justify your answer.

$\cot$ $A$ is not defined for $A =0^{\circ}$

Express the ratios $\cos A ,$ tan $A$ and $\sec A$ in terms of $\sin A .$

Prove that

$\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}=\frac{1}{\sec \theta-\tan \theta},$ using the identity

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