Note that .
So find the Taylor series for , and integrate it term by term to find the Taylor series for .
tan(x+pi/4) cos(x + pi/4) = sin(x + pi/4)
tan(x+pi/4) * 0.5 *root(2) (cos(x) - sin(x)) = 0.5 * root(2) (sin(x) + cos(x))
tan(x+pi/4) (cos(x) - sin(x)) = sin(x) + cos(x)
I then let tan(x+pi/4) = a + bx + cx^2 , subbed in the expansions for sin(x) and cos(x) and equated coefficients. Again, can you see an improvement upon this method? Time is very precious in the exam so I'm trying to optimise my working.