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

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

$\sin \theta=\cos \theta$ for all values of $\theta$

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

$(\operatorname{cosec} \theta-\cot \theta)^{2}=\frac{1-\cos \theta}{1+\cos \theta}$

Consider $\triangle ACB$, right-angled at $C$, in which $AB =29$ units, $BC =21$ units and $\angle ABC =\theta$ (see $Fig.$). Determine the values of

$(i)$ $\cos ^{2} \theta+\sin ^{2} \theta$

$(ii)$ $\cos ^{2} \theta-\sin ^{2} \theta$

Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$

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