trigonometry - Please prove $frac{1 + sintheta - costheta}{1 + sintheta + costheta} = tan left(frac theta 2right)$

Prove that $\dfrac{1 + \sin\theta - \cos\theta}{1 + \sin\theta + \cos\theta} = \tan\left(\dfrac{\theta}{2}\right)$

Also it is a question of S.L. Loney's Plane Trignonometry

What I've tried by now:

$$\begin{align} & =\frac{1+\sin\theta-\sin(90-\theta)}{1+\cos(90-\theta)+\cos\theta} \\[10pt] & =\frac{1+2\cos45^\circ \sin(\theta-45^\circ)}{1+2\cos45^\circ \cos(45-\theta)} \end{align}$$

Cause I do know
$$\begin{align} & \sin c + \sin d = 2\sin\left(\frac{c+d}{2}\right)\cos\left(\frac{c-d}{2}\right) \\[10pt] \text{and } & \cos c + \cos d = 2\cos\left(\frac{c+d}{2}\right)\sin\left(\frac{c-d}{2}\right) \end{align}$$

I can't think of what to do next..

Answer

Let $\theta = 2 \phi$, then the thing to be proven is:

Prove that $$\frac{1 + \sin(2\phi) - \cos(2\phi)}{1 + \sin(2\phi) + \cos(2\phi)} = \tan(\phi)$$

Then use:

$$\sin(2\phi) = 2 \sin \phi \cos \phi$$
$$\cos(2\phi) = \cos^2 \phi - \sin^2 \phi$$

and:

$$\sin^2 \phi + \cos^2 \phi = 1$$