Question

Cos^2x-sin^2x=2cosx-1

Answer 1

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Answer 2

To solve the equation cos^2x - sin^2x = 2cosx - 1, we can use the trigonometric identity cos^2x - sin^2x = cos(2x).

Substituting this identity into the equation, we have:

cos(2x) = 2cosx - 1

Now, let's simplify the equation:

2cos^2x - 1 = 2cosx - 1

Rearranging the terms:

2cos^2x - 2cosx = 0

Factoring out 2cosx:

2cosx(cosx - 1) = 0

Now we can apply the zero product property, which states that if a product of factors is equal to zero, then at least one of the factors must be zero.

Setting each factor to zero and solving for x:

1. 2cosx = 0 cosx = 0

2. cosx - 1 = 0 cosx = 1

Now, let's find the solutions for x in the range of 0 to 2π (or 0 to 360 degrees).

1. cosx = 0: The solutions for cosx = 0 in this range are x = π/2 (90 degrees) and x = 3π/2 (270 degrees).

2. cosx = 1: The solution for cosx = 1 in this range is x = 0 (0 degrees).

Therefore, the solutions to the equation cos^2x - sin^2x = 2cosx - 1 in the given range are x = 0, π/2, and 3π/2.