Write the given expression as the tangent of an angle.
tan 7x − tan x/ 1 + tan 7x tan x
I am sure you meant:
(tan 7x − tan x)/( 1 + tan 7x tan x)
compare to :
tan(A-B) = (tanA - tanB)/(1 + tanAtanB)
so would that be tan (8x)?
Not unless 7-1 = 8
sorry I did 7+1
I guess that's what they did too as I had the same question and made the same mistake
Well, this expression seems to be quite tangential, doesn't it?
To rewrite this expression as the tangent of an angle, we need to apply a trigonometric identity called the tangent of a difference formula. The formula is as follows:
tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
Looking at our given expression, we have tan 7x − tan x in the numerator and 1 + tan 7x tan x in the denominator.
Now, to utilize the tangent of a difference formula, we need to let A = 7x and B = x. Replacing A and B in the formula, we get:
tan(7x - x) = (tan 7x - tan x) / (1 + tan 7x tan x)
Simplifying the expression, we have:
tan(6x) = (tan 7x - tan x) / (1 + tan 7x tan x)
So, the given expression can be rewritten as the tangent of an angle: tan(6x). Voilà!
To write the given expression as the tangent of an angle, we can use the trigonometric identity for the difference of tangents:
tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
Comparing this identity to the given expression, we can see that A = 7x and B = x.
Substituting these values into the identity, we have:
tan(7x - x) = (tan 7x - tan x) / (1 + tan 7x tan x)
Therefore, the given expression can be written as the tangent of an angle as:
tan(6x) = tan 7x - tan x / 1 + tan 7x tan x