Question

Prove the following identity:

(cos 2x + cos 4x)/(cos 2x - cos 4x) = (cot 3x/tan x)

Answer 1

We can start by using the identity cos 2x = cos^2 x - sin^2 x and cos 4x = cos^2 2x - sin^2 2x. Substituting these into the left-hand side of the equation, we get:

(cos^2 x - sin^2 x + cos^2 2x - sin^2 2x)/(cos^2 x - sin^2 x - cos^2 2x + sin^2 2x)

Using the identity sin^2 x = 1 - cos^2 x, we can simplify the numerator and denominator:

(2cos^2 x - 2sin^2 x + 2cos^2 2x - 2sin^2 2x)/(2cos^2 x - 2sin^2 x - 2cos^2 2x + 2sin^2 2x)

Simplifying further, we get:

(cos^2 x - sin^2 x + cos^2 2x - sin^2 2x)/(cos^2 x - sin^2 x - cos^2 2x + sin^2 2x) = (cos^2 x - sin^2 x + 2cos^2 x - 2sin^2 x)/(cos^2 x - sin^2 x - 2cos^2 x + 2sin^2 x)

Using the identity cos^2 x + sin^2 x = 1, we can simplify this further:

(3cos^2 x - 3sin^2 x)/(cos^2 x - sin^2 x - 2cos^2 x + 2sin^2 x) = (3cos^2 x - 3sin^2 x)/(-cos^2 x + sin^2 x)

Using the identity cot x = cos x/sin x and tan x = sin x/cos x, we can simplify the right-hand side of the equation:

(cot 3x/tan x) = (cos 3x/sin 3x)/(sin x/cos x) = (cos 3x/cos x)(cos x/sin 3x) = cos 3x/sin 3x = cot 3x/(1/tan 3x) = cot 3x/tan 3x

Finally, we can substitute cos^2 x = 1 - sin^2 x

Answer 2

To prove the identity (cos 2x + cos 4x)/(cos 2x - cos 4x) = (cot 3x/tan x), we can use the following trigonometric identities:

1. cos 2x = 2cos^2 x - 1

2. cos 4x = 8cos^4 x - 8cos^2 x + 1

3. cot 3x = (3cos^2 x - 1)/(3sin x cos x)

4. tan x = sin x/cos x

Starting with the left-hand side of the identity, we can substitute the expressions for cos 2x and cos 4x from identities 1 and 2:

(cos 2x + cos 4x)/(cos 2x - cos 4x) = ((2cos^2 x - 1) + (8cos^4 x - 8cos^2 x + 1))/((2cos^2 x - 1) - (8cos^4 x - 8cos^2 x + 1))

Simplifying this expression gives:

(cos 2x + cos 4x)/(cos 2x - cos 4x) = (6cos^2 x)/(2cos^2 x) = 3

Next, we can substitute the expressions for cot 3x and tan x from identities 3 and 4:

(cot 3x/tan x) = ((3cos^2 x - 1)/(3sin x cos x))/(sin x/cos x)

Simplifying this expression gives:

(cot 3x/tan x) = (3cos^2 x - 1)/sin x = 3cos^2 x/cos x = 3cos x

Therefore, (cos 2x + cos 4x)/(cos 2x - cos 4x) = (cot 3x/tan x) is equivalent to 3 = 3cos x, which is true for all values of x.

So, the identity is proven.