How to Solve cos(x+y)dy=dx Using Trigonometric Identities?

[abrowaqas]

cos(x+y)dy=dx

I have tried to solve it by
Letting
t = x+y

But it's not going to be separated.. Some one please help..

 

[Eynstone]

Consider dx/dy. The equation becomes (1 +cost)=dt/dy.

 

[HallsofIvy]

I don't see why you would say it does not become separable. If t= x+ y then dt= dx+ dy so dy= dt- dx. cos(x+y)dy= dx becomes cos(t)(dt- dx)= cos(t)dt- cos(t)dx= dx or cos(t)dt= (1+ cos(t))dx.
$$dx= \frac{cos(t)}{1+ cos(t)}dt$$.

 

[abrowaqas]

I got the answer ..
Let u=x+y
du/dx=1+dy/dx
dy/dx=du/dx-1
Applying this:
cos(x+y)dy=dx
cos(x+y)dy/dx=1
cosu(du/dx-1)=1
du/dx-1=sec u
du/dx=1+sec u
dx/du=1/(1+sec u)
x=integral of 1/(1+sec u) du

 

[blue_raver22]

I would suggest using the trigonometric identity cos(x+y) = cos(x)cos(y) - sin(x)sin(y) to rewrite the equation as:

cos(x)cos(y)dy - sin(x)sin(y)dy = dx

Then, we can let u = cos(y) and du = -sin(y)dy to get:

cos(x)du = dx

This equation can now be separated and solved by integrating both sides with respect to their respective variables. This approach utilizes the properties of trigonometric identities to simplify the equation and make it more manageable to solve. Additionally, it is important to carefully choose substitutions and transformations in order to effectively separate variables and find a solution.