Let  pi = 3.1419926..., as usual.  
Think of a right triangle with angles 
pi/2 (of course), x, and u.  
Then, since the sum of all the angles 
in any triangle is equal to pi, 
x and u are related by 
    x + u = pi/2 
or, equivalently, 
    u = pi/2 - x .
Note also that, in this triangle, 
we have, by the very definitions of sin and cos, 
    cos(x) = sin(u) 
and, similarly, 
    sin(x) = cos(u) .
If we use the relation between x and u 
that we noticed above, we see that 
    cos(x) = sin(pi/2-x).  
So far we proved this relation only for angles 
x between 0 and pi/2 (why?), but one can show 
that it is true for any angle x.  

Now look at the integral in the left-hand side 
of the identity given in the problem.  
Make the substitution 
    u = pi/2-x 
and use the relation 
    cos(x) = sin(pi/2-x) 
to obtain the desired result.