The area of the region above the x-axis bounded by the curve y=tanx, 0≤x≤π/2 and the tangent to the curve at x=π/4 is :

y=tanx
dy/dx=sec²x=sec²(π/4)=2.
y−1=2(x−π/4)
y=2x−π/2+1
At y=0, 2x=π/2 ⇒x=(π/4)
A1=∫₀^(π/4)tanxdx=[log|secx|]₀^(π/4)=log|sec(π/4)|−log|sec(0)|=log√2
A2=∫π/4^(π/4)(tan⁻¹(2x−π/2+1))dx=1/2−log(1/√2)−log|cos(π/4)|+π/8(π/4)(1/2)
A1+A2=1/2(log2)