$9 \sec ^{2} A-9 \tan ^{2} A=..........$

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

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

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

If $\sec 4 A =\operatorname{cosec}\left( A -20^{\circ}\right),$ where $4 A$ is an acute angle, find the value of $A$. (in $^{\circ}$)

Evaluate the following:

$\frac{\cos 45^{\circ}}{\sec 30^{\circ}+\operatorname{cosec} 30^{\circ}}$

Prove that

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

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

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