Right option is (a) 2(1-cos\frac{1}{\sqrt{2}})
For explanation I would say: Let I=\int_{π/4}^{π/2} \,2sinx \,sin(cosx) \,dx
F(x)=\int 2 \,sinx \,sin(cosx)dx
Let cosx=t
Differentiating w.r.t x, we get
sinx dx=dt
∴\int 2 \,sinx \,sin(cosx)dx=\int 2 \,sint \,dt=-2 \,cost
Replacing t with cosx, we get
∴∫ 2 sinx sin(cosx)dx=-2 cos(cosx)
By applying the limits, we get
I=F(\frac{π}{4})-F(\frac{π}{2})=-2 cos(\frac{cosπ}{4})+2 cos(\frac{cosπ}{2})
=2(1-cos\frac{1}{\sqrt{2}})