Proving Trig Problem: sec (2x) - 1 = sin^2 x / 2 sec (2x)

[Sixlets]

I'm stuck :/ I have to prove the following:
sec (2x) - 1 = sin^2 x
____________
2 sec (2x)

Unfortunately, I don't know any terms that are equal to sec (2x). Any help would be awesome!

 

[Eldar]

$$sec(2x)=\frac{1}{cos(2x)}=\frac{1}{cos^2x-sin^2x}$$

Also you can change the denominator by substituting in for $$cos^2x$$ or $$sin^2x$$.

 

[QuarkCharmer]

I think I would start with the left side of the eq. and use the double angle identity to get it in terms of sin/cos.

 

[Sixlets]

I'll try that again. I kept getting way off track, but I probably just got confused with all the fractions haha. Thanks everyone!

 

[CellCoree]

Hello,

I understand that you are trying to prove the equation sec (2x) - 1 = sin^2 x / 2 sec (2x) and are looking for guidance on how to proceed. Let's break down the equation and see if we can simplify it.

First, let's rewrite the equation as:

(sec (2x) - 1)/2sec (2x) = sin^2 x

We can then use the identity sec x = 1/cos x to rewrite the left side as:

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

Next, we can simplify the numerator by finding a common denominator:

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

Using the identity cos (2x) = 1 - 2sin^2 x, we can substitute this into the equation to get:

(1 - (1 - 2sin^2 x))/2(1 - 2sin^2 x) = sin^2 x

Simplifying further, we get:

2sin^2 x/2(1 - 2sin^2 x) = sin^2 x

Cancelling out the 2's on the left side, we are left with:

sin^2 x/(1 - 2sin^2 x) = sin^2 x

Using the identity sin^2 x = 1 - cos^2 x, we can substitute this into the equation to get:

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

Simplifying further, we get:

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

Combining like terms, we get:

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

Finally, using the identity sin^2 x = 1 - cos^2 x, we can substitute this into the equation to get:

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

And this is true, since both sides are equal to 1 - cos^2 x.

I hope this helps you understand the steps to prove the given equation