Prove the identity. (show your work please)
tan(x − π/4)= (tan x − 1)/(tan x + 1)
Thank you!
LS = (tanx - tan(π/4)/(1 + tanx(tan(π/4)) )
= (tanx - 1)/(1 + tanx)
= RS
I used the identity
tan(A-B)= (tanA - tanB)/(1 + tanAtanB)
and the fact that tan π/4 = tan 45°= 1
To prove the given trigonometric identity, we need to manipulate one side of the equation until it matches the other side. Here's how we can do it step by step:
1. Start with the left side of the equation: tan(x - π/4).
2. Use the trigonometric identity for the difference of angles: tan(a - b) = (tan a - tan b) / (1 + tan a * tan b).
Applying this identity, we have: tan(x - π/4) = (tan x - tan (π/4)) / (1 + tan x * tan (π/4)).
3. Recall that tan (π/4) = 1. Substituting this value, we get: tan(x - π/4) = (tan x - 1) / (1 + tan x * 1).
4. Simplify the denominator: tan(x - π/4) = (tan x - 1) / (1 + tan x).
5. Now, compare the simplified left side with the right side of the equation: (tan x - 1)/(tan x + 1).
Therefore, we have successfully proven the identity:
tan(x - π/4) = (tan x - 1) / (tan x + 1).