Where do I start to prove this identity:

Question

Where do I start to prove this identity:

sinx/cosx= 1-cos2x/sin2x

please help!!

Hint: Fractions are evil. Get rid of them.

Well, cos2x = cos²x - sin²x, so
1-coscx = 1 - cos²x - sin²x =
1 - cos²x + sin²x
You should be able to simplify this to 2*something squared.
The denominator is sin2x = 2sin(x)cos(x)
You should be able to finish this, if not post a question.

The second line should be
1 - cos2x = 1 - (cos²x - sin²x)
then the 3rd line will make sense.
1 - cos²x should look familiar.

Answer 1

To prove the given identity: sinx/cosx = (1 - cos2x)/sin2x

First, let's simplify the right side of the equation using the given hints.

cos2x = cos^2x - sin^2x (This is an identity called the double angle identity for cosine)

Now let's substitute this into the equation:

(1 - cos2x)/sin2x = (1 - (cos^2x - sin^2x))/(2sinxcosx)

Simplifying further, we get:

= (1 - cos^2x + sin^2x)/(2sinxcosx)

Now, we know that sin^2x + cos^2x = 1 (This is the Pythagorean identity for trigonometric functions), so we can substitute:

= (1 - cos^2x + (1 - cos^2x))/(2sinxcosx)

= (2 - 2cos^2x)/(2sinxcosx)

Next, we can factor out a 2 in the numerator:

= 2(1 - cos^2x)/(2sinxcosx)

The term (1 - cos^2x) can be recognized as sin^2x, so we can substitute:

= 2sin^2x/(2sinxcosx)

The term 2sinx can cancel out in the numerator and denominator, leaving us with:

= sinx/cosx

Thus, we have proven that sinx/cosx = (1 - cos2x)/sin2x.

Remember, when solving trigonometric identities, it's important to know the trigonometric identities and rules, such as the Pythagorean identity and double angle identity, and use them appropriately to simplify the expression.