# Question:Better (real) solutions to an equation involving trigonometric functions?

## Question:Better (real) solutions to an equation involving trigonometric functions?

Maple 2023

For example, here are two equations containing trigonometric functions (Note that they do not form one system!):

restart; # There are more examples, yet for the sake of briefness, they are omitted here.
eqn__0 := cos(x)*cos(y)*cos(x + y) = 2*(sin(x)*sin(y) - 1)*2*(sin(x)*sin(x + y) - 1)*2*(sin(y)*sin(x + y) - 1):
eqn__1 := (cos(x + y) - (cos(x) + cos(y)) + 1)**2 + 2*cos(x)*cos(y)*cos(x + y) = 0:

Unfortunately, none of

(* Tag０ *) RealDomain:-solve(eqn__0, {y, x}):
(* Tag１ *) solve(eqn__0, {y, x}) assuming y + x >= 0, (y, x) <=~ Pi:
(* Tag２ *) RealDomain:-solve(eqn__1, {y, x}):
(* Tag３ *) solve(eqn__1, {y, x}) assuming y + x >= 0, (y, x) <=~ Pi:

outputs concise solutions.
Using `plot3d`, it is easy to check that when "And(y + x >= 0, (y, x) <=~ Pi)", “{y = Pi/2, x = 0}, {y = Pi/3, x = Pi/3}, {y = 0, x = Pi/2}, {y = Pi/2, x = Pi/2}” is both the only solution to "eqn__0" and the only solution to "eqn__1". But how to get Maple to do so without manual intervention?

Edit. The main purpose is to automatically find the generic solutions to each of the two equations (Tag０ and Tag２) (separately). Now that the cosine and sine functions are both periodic with period 2π and both (lhs - rhs)(eqn__0) and (lhs - rhs)(eqn__1) are even symmetric, it is enough to focus only on the region y + x ≥ 0 ∧ (y, x) ≤~ Pi. So, in theory, a second-best workaround should be Tag１ and Tag３. However, why is Maple still unable to find the four exact solutions above?

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