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)

set the right sides equal and multiply by LCD to get

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

x= 0 is an obvious solution

x= 90 is also a solution as sin(135)cos-45 = sin-45cos135

= sqr2/2)(sqr2/2) = -sqr2/2)(-sqr2/2)

= 2/4 = 2/4

there are an infinite number of solutions

-pi, -pi/2, 0, pi/2, pi, 3pi/2, 2pi. etc

or simply pin/2 where n = any integer

or 90n degrees, where n = any integer

solve by graphing and see where they intersect

or try the sum/difference formulas

tan(45+x) = (tan 45 + tanx)/(1-tan45tanx) = (tan45- tanx)/(1+tan45tanx)

= (1+tanx)/(1-tanx) = (1- tanx)/(1+tanx)

cross multiply

1+2tanx + tan^2(x) = 1 -2tanx + tan^2(x)

4tanx=0

tanx = 0

x = tan^-1(0) = pin where n = any integer

or 180n degrees where n = any integer

90 or pi/2 makes the original equation in sum/difference fractional form undefined and indeterminant = infinity/-infinity = -infinity/infinity which could be true, but may not be

possibly analyze the limit as x approaches 90 with L'Hopitals rule

that gives sec^2(x)/2tanxsec^2(x) = 1/2tanx

plug x= 0 and you get 1/0

keep trying more derivatives until you get a real number limit

Answer 3

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 4

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