Express as a single sine or cosine function (note: this is using double angle formulas)

g) 8sin^2x-4
I just don't get this one. I know it's got something to do with the 1-2sin^2x double angle formula. It's the opposite though? :S

h) 1-2sin^2 (π/4-x/2)
= 1-sin^2(π/4-x/2)-sin^2(π/4-x/2)
= cos^2(π/4-x/2)-sin^2(π/4-x/2)
I got all the way up to cos (π/4 - x/2 + π/4 - x/2)

The answer is supposed to be sin x. I have no clue how they got that.

g) 8sin^2x-4

= 4(2sin^2 x - 1)
= 4(-cos 2x)
= -4cos 2x

h) 1-2sin^2 (π/4-x/2)
= cos 2(π/4-x/2)
= cos (π/2-x)
= cos(π/2)cosx + sin(π/2)sinx
= 0(sinx) + 1(sinx)
= sinx

Thanks a lot!

But can you please explain how you got 4(2sin^2 x - 1) for g and cos 2(π/4-x/2) for h?

I think I might understand h because 1-sin^2x = cosx but wouldn't it just be cos(π/4-x/2) cos(π/4-x/2)?

"But can you please explain how you got 4(2sin^2 x - 1)"

I took out a common factor of 4

"..and cos 2(π/4-x/2) for h"

Ok, let's work it in reverse.
You know that cos 2A = cos^2 A - sin^2 A, or cos 2A = 1 - 2sin^2 A ,right?

so I simply let A = (π/4-x/2)
then 2A = 2(π/4-x/2)
= π/2 - x

To express the given expression (8sin^2x - 4) as a single sine or cosine function, we can make use of the double angle formulas.

Here's how you can approach it:

g) 8sin^2x - 4

Step 1: Recall the double angle identity for sine: sin(2θ) = 2sinθcosθ.

Step 2: Rewrite sin^2x as (1 - cos^2x). This can be done using the Pythagorean identity: sin^2x + cos^2x = 1.

Step 3: Substitute the double angle identity and the expression from step 2 into the original equation.

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

Step 4: Now, we can express -8cos^2x + 4 as a single cosine function using the identity: cos(2θ) = 2cos^2θ - 1.

Using this identity, we can rewrite the expression as:

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

So, the expression 8sin^2x - 4 can be expressed as -8cos(2x) + 4.

Moving on to part h:

h) 1 - 2sin^2(π/4 - x/2)

Step 1: Expand sin^2(π/4 - x/2) using the identity sin^2θ = 1 - cos^2θ.

1 - 2sin^2(π/4 - x/2) = 1 - 2(1 - cos^2(π/4 - x/2))

Step 2: Simplify the expression further.

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

Step 3: Use the double angle identity for cosine: cos(2θ) = 2cos^2θ - 1.

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

Step 4: Simplify further.

-1 + cos(2(π/4 - x/2)) = cos(π/2 - x) = sin(x)

Thus, the expression 1 - 2sin^2(π/4 - x/2) can be simplified to sin(x).