Let f(x)  be an even function, I1=∫0π2f(cos2x)cosxdx,I2=∫0π4f(sin2x)cosxdx . Then I1I2=

I1=∫0π4f(cos2x)cosxdx+∫π4π2f(cos2x)cosxdx Setting x=π4−t  in the first integral and x=π4+t  in the second integral,I1=∫0π4f(sin2t)cos(π4−t)dt+∫0π4f(sin2t)cos(π4+t)dt =∫0π4f(sin2t)[cos(π4−t)+cos(π4+t)]dt =2∫0π4f(sin2t)costdt=2I2