Answer
By multiplying both numerator and denominator by $(1-\cos A)$, we can derive the equivalent identity as shown below.
Work Step by Step
$$\tan\frac{A}{2}=\frac{\sin A}{1+\cos A}$$
The job is to derive the equivalent identity:
$$\tan\frac{A}{2}=\frac{1-\cos A}{\sin A}$$
As hinted by the exercise, we multiply both numerator and denominator of the original identity by $(1-\cos A)$
$$\tan\frac{A}{2}=\frac{\sin A(1-\cos A)}{(1+\cos A)(1-\cos A)}$$
- Denominator: $$(1+\cos A)(1-\cos A)=1-\cos^2A$$ (as $(A+B)(A-B)=A^2-B^2$)
$$(1+\cos A)(1-\cos A)=\sin^2A$$ (Pythagorean Identity)
Therefore,
$$\tan\frac{A}{2}=\frac{\sin A(1-\cos A)}{\sin^2 A}$$
$$\tan\frac{A}{2}=\frac{1-\cos A}{\sin A}$$
which is the equivalent identity.
Thus, we have completed deriving the equivalent identity as requested.
