Prove each identity:

a) 1-cos^2x=tan^2xcos^2x
b) cos^2x + 2sin^2x-1 = sin^2x

I also tried a question on my own:
tan^2x = (1 – cos^2x)/cos^2x
R.S.= sin^2x/cos^2x
I know that the Pythagorean for that is sin^2x + cos^2x
That's all I could do.

a)

Given tan(θ)=sin(θ)/cos(θ)
substitute tan(θ) by sin(θ)/cos(θ) on the RHS and see what you get.

b) hint:
cos(2θ)=cos²(θ)-sin²(θ)

To prove each identity, we will work with one side of the equation at a time and simplify it until it matches the other side.

a) 1 - cos^2(x) = tan^2(x) cos^2(x)

To start, we will manipulate the left-hand side (LHS):
1 - cos^2(x) = sin^2(x) (since 1 - cos^2(x) = sin^2(x) according to the Pythagorean identity)

Now, let's manipulate the right-hand side (RHS):
tan^2(x) = sin^2(x) / cos^2(x) (using the definition of tangent: tan(x) = sin(x) / cos(x))

Now we have:
LHS = sin^2(x)
RHS = sin^2(x) / cos^2(x)

Next, let's simplify the RHS:
RHS = sin^2(x) / cos^2(x)
= sin^2(x) * (1 / cos^2(x))
= sin^2(x) * sec^2(x) (since the reciprocal of cosine is secant, sec(x) = 1 / cos(x))

So now we have:
LHS = sin^2(x)
RHS = sin^2(x) * sec^2(x)

Using the identity sec^2(x) = 1 + tan^2(x), we can rewrite the RHS:
RHS = sin^2(x) * (1 + tan^2(x))

Finally, we can see that LHS = RHS:
sin^2(x) = sin^2(x) * (1 + tan^2(x))
This identity is true, and we have proven it.

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

We will manipulate the left-hand side (LHS) to match the right-hand side (RHS):

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

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can substitute:

LHS = (1 - sin^2(x)) + 2sin^2(x) - 1

Simplifying the expression:

LHS = 1 - sin^2(x) + 2sin^2(x) - 1
= - sin^2(x) + 2sin^2(x)

Combining like terms:

LHS = sin^2(x)

Now we can see that LHS = RHS, since they are both sin^2(x). Hence, the identity cos^2(x) + 2sin^2(x) - 1 = sin^2(x) is true.

Regarding your own question:

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

To prove this identity, you can follow a similar process as shown above for the first identity. Use the Pythagorean identity sin^2(x) + cos^2(x) = 1 and manipulate both sides of the equation until they match. Specifically, using the Pythagorean identity, you can substitute sin^2(x) for 1 - cos^2(x), simplifying the expression further and showing that both sides are equal.