1.) Find the solution set of

2sin squared θ-1= 0 if 180 degrees is less than or equal to θ is less than or equal to 360 degrees

2.) prove:

2cotθ/cot squared θ -1 = sin2 θ sec2 θ

2 sin^2 Ø = 1

sin^2 Ø = 1/2
sin Ø = 1/√2
if 180 ≤ Ø ≤ 360°
then Ø = 225° or 315°

2.

Did you mean:
2cotØ/(cot^2 Ø - 1) = sin^2 Ø sec^2 Ø ?
If so, then it is not and identity, (I tried it with Ø=20°)
did you mean:
2cotØ/cot^2 Ø - 1 = sin^2 Ø sec^2 Ø ?
If so, then it is not and identity, (I tried it with Ø=20°)

did you mean:
2cotØ/cot^2 Ø - 1 = sin 2Ø sec 2Ø ?
If so, then it is not and identity, (I tried it with Ø=20°)

etc.

4^(x 1) - 6^x - 2.9^(x 1) = 0

1.) To find the solution set of the equation 2sin^2θ - 1 = 0, we can use some trigonometric identities and algebraic manipulation.

First, let's rewrite the equation in terms of sin^2θ:
2sin^2θ = 1

Then, divide both sides by 2:
sin^2θ = 1/2

Taking the square root of both sides:
sinθ = ±√(1/2)

Since we are given that 180 degrees ≤ θ ≤ 360 degrees, we need to find angles within that range whose sine is equal to ±√(1/2).

The principal angle whose sine is √(1/2) is 45 degrees (or π/4 radians). However, we want to consider angles within the given range, so we need to find the reference angles that fall within that range.

In the second quadrant (180 ≤ θ ≤ 270), the reference angle for the sine of √(1/2) is 180 - 45 = 135 degrees (or 3π/4 radians). In the third quadrant (270 ≤ θ ≤ 360), the reference angle is 360 - 45 = 315 degrees (or 7π/4 radians).

Therefore, the solution set for the equation 2sin^2θ - 1 = 0, within the given range, is:
θ = 135 degrees, 315 degrees (or 3π/4, 7π/4 radians).

2.) To prove the equation 2cotθ/cot^2θ - 1 = sin^2θ sec^2θ, we need to simplify both sides of the equation using trigonometric identities and show that they are equal.

Starting with the left side of the equation:
2cotθ/cot^2θ - 1
Using the identity cot^2θ = 1 + tan^2θ:
2cotθ/(1 + tan^2θ) - 1
Since cotθ = 1/tanθ:
2(1/tanθ)/(1 + tan^2θ) - 1
Simplifying the expression on the numerator:
2/tanθ / (1 + tan^2θ) - 1
Now, find a common denominator for the fractions:
(2 - tan^2θ) / (tanθ(1 + tan^2θ)) - 1

Moving on to the right side of the equation:
sin^2θ sec^2θ
Since secθ = 1/cosθ, sec^2θ = 1/cos^2θ:
sin^2θ / (1/cos^2θ)
Multiplying the numerator and denominator by cos^2θ:
sin^2θ * cos^2θ / 1
sin^2θ * cos^2θ

To compare both sides of the equation, we will rewrite the expression on the right side using trigonometric identities:
sin^2θ * cos^2θ
Using the identity sin^2θ = 1 - cos^2θ:
(1 - cos^2θ) * cos^2θ
Expanding and simplifying:
cos^2θ - cos^4θ

Now we can see that the left side and right side of the equation are equal:
(2 - tan^2θ) / (tanθ(1 + tan^2θ)) - 1 = cos^2θ - cos^4θ

Thus, the equation 2cotθ/cot^2θ - 1 = sin^2θ sec^2θ is proven.