Introduction to Trigonometry

What is the value of $\frac{1 - \tan^2 45^\circ}{1 + \tan^2 45^\circ}$?

If $\sin A = \frac{3}{4}$, then what is the value of $\cos A$?

If $\sin A = \frac{3}{4}$, calculate $\cos A$ and $\tan A$. Show all steps.

Prove the trigonometric identity: $\frac{\sin \theta - 2 \sin^3 \theta}{2 \cos^3 \theta - \cos \theta} = \tan \theta$. Show all steps clearly.

Prove the identity: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$. Show all steps clearly.