tan (45+x) = tan (45-x). when using compound formula x=90 disappears

Question

When solving by simply making 45+x=45-x you also get x=90. My maths teacher has gone over and checked on the actual graph and this is definetly a valid answer, we can't figure out why it doesnt work when using compound angle formula though?

Answer 1

This is a very interesting question!

The solution x = 90 degrees cannot be obtained by solving an equation for tan(x) because tan(90 degrees) is not defined, strictly speaking. So you cannot obtain that solution by solving for tan(x) to be a particular value.

The correct method to solve that equation is to realize that if tan(A) = tan(B), then A and B differ by n*pi, where n is an integer.

Answer 2

tan(45+x)=sin(45+x)/cos(45+x)

tan(45-x)=sin(45-x)/cos(45-x)

Let the sin/cos forms equal each other:

sin(45+x)/cos(45+x)=sin(45-x)/cos(45-x)

sin(45+x)*cos(45-x)=cos(45+x)*sin(45-x)

If you use the double angle.formula, simplify terms, and solve for x, then 90 degrees will be one of the solutions

Answer 3

Using the formula requires the implicit assumption that x = π/2 + nπ/2 since the tangent of those angles is undefined.