I tried to solve it and cant get the right answer please hel

I tried to solve it and cant get the right answer please help!

Solve the given differential equation by using an appropriate substitution. The DE is of the form dy/dx = f(Ax + By + C),

which is given in (5) of Section 2.5.

dy/dx= tan²(x + y)

dy/dx= tan2(x + y)……………………………….1

The DE is in form of dy/dx = f(Ax + By + C)

So Let

U=x+y

y = U-x

Taking derivative on both sides w.r.t x

dy/dx = du/dx – 1

Putting value of y in eq 1

du/dx -1 = tan2 (u)

du/dx = tan2 (u) + 1

= sec2 (u)…………………..[ tan2 (u) + 1= sec2 (u]

du/sec2 (u) = dx

cos2 (u) . du = dx………………………[1/sec2 (u) =cos2 (u)]

Taking integration on both sides

cos2 (u) . du = dx

By double angle rule we know that

Cos2 (x) = [cos (2x)+1]/2

Putting value in above

{ [cos (2x)+1]/2}.du= dx

(1/2) du + (1/2)cos (2u) du = dx

(1/2)u + (1/2). [sin(2u)/2] = x+c1

use u=x+y

(1/2).(x+y) +(1/4).sin[2(x+y)] = x +c1

Multiply by 4

2(x+y) +sin(2x+2y) = 4x +4c1

2y-2x+ sin(2x+2y=4c1=C………………………..C= 4c1