EQUATION OR IDENTITY? tany−cotx=cos(y−x)/ sinxcosy

Question

 tany−cotx=cos(y−x)/ sinxcosy

can this just be answered by plugging in random values for x and y and checking if LHS=RHS

Answer 1

If you are asked to prove a given identity, then question has to be answered algebraically usually aided by already proven trigonometric relationships such as

1. known fundamental trigonometric identities (for example, sin²x + cos²x = 1)
2. known trigonometric formulas (e.g, sum and difference formulas)

If you are asked to determine whether or not a given equation is an identity, this can also be done algebraically. For example, if you can prove algebraically that LHS ≠ RHE then you have proven that that the equation is not an identity.

In some cases its also possible to make this determination by plugging values of x and y. For example, if you can find any pair of x and y values where LHS ≠ RHE this also proves the equation is not an identity.

However, if every pair of x and y values you try results in LHS = RHE this doesn't actually prove an identity since there may an untried pair of x and y values where LHS ≠ RHE.

If you can't disprove an identity by trial and error, then you have to prove it (or disprove it) algebraically.

Maybe a little long winded, but I hope this helps.

Answer 2

tany - cotx = siny/cosy - cosx/sinx = (sinxsiny - cosxcosy)/sinxcosy = - cos(x + y)/(sinxcosy)