Where is this double angle from (2Cos^2X-1)
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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The double angle identity $\cos 2x = 2\cos^2 x - 1$ is derived from the cosine addition formula and the Pythagorean identity.
Step 1: Start with the cosine addition formula.
$$ \cos(A+B) = \cos A \cos B - \sin A \sin B $$
Step 2: Let $A=x$ and $B=x$.
$$ \cos(x+x) = \cos x \cos x - \sin x \sin x $$
$$ \cos 2x = \cos^2 x - \sin^2 x $$
Step 3: Use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$, which means $\sin^2 x = 1 - \cos^2 x$.
Substitute this into the equation from Step 2:
$$ \cos 2x = \cos^2 x - (1 - \cos^2 x) $$
Step 4: Simplify the expression.
$$ \cos 2x = \cos^2 x - 1 + \cos^2 x $$
$$ \cos 2x = 2\cos^2 x - 1 $$
This shows how the identity $\cos 2x = 2\cos^2 x - 1$ is derived.
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