Q.Show that tan(π/4 + (1/2)cos⁻¹(a/b)) + tan(π/4 - (1/2)cos⁻¹(a/b)) = 2b/a.

Expand both tangents using the standard $\tan(\pi/4\pm x)$ formulas in terms of $t=\tan\left(\tfrac12\cos^{-1}\tfrac ab\right)$, then use the half-angle cosine formula to simplify.

Step 1. Let $\theta=\cos^{-1}\dfrac ab$, so $\cos\theta=\dfrac ab$. Let $t=\tan\dfrac\theta2$.

Step 2. Use $\tan\!\left(\dfrac\pi4+x\right)=\dfrac{1+\tan x}{1-\tan x}$ and $\tan\!\left(\dfrac\pi4-x\right)=\dfrac{1-\tan x}{1+\tan x}$ with $x=\theta/2$:

$\tan\!\left(\frac\pi4+\frac\theta2\right)+\tan\!\left(\frac\pi4-\frac\theta2\right) = \frac{1+t}{1-t}+\frac{1-t}{1+t} = \frac{(1+t)^2+(1-t)^2}{1-t^2} = \frac{2(1+t^2)}{1-t^2}.$

…