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Prove the identity.
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Q. Prove the identity.
- Apply Double Angle Formulas: We will start by using the double angle formulas for sine and cosine. The double angle formula for sine is , and for cosine is . We will apply these formulas to the left side of the identity.
- Rewrite : Rewrite using the double angle formula:.
- Rewrite : Rewrite using the double angle formula and the Pythagorean identity :.
- Substitute into Identity: Now, we substitute the expressions we found into the left side of the identity: .
- Simplify Denominator: Since , we can simplify the denominator:$(\(2\) \sin x \cos x) / (\cos^\(2\) x + \sin^\(2\) x) = (\(2\) \sin x \cos x) / \(1\) = \(2\) \sin x \cos x.
- Divide by \(\cos x\): Now, we divide both the numerator and the denominator by \(\cos x\), assuming \(\cos x \neq 0\) (since division by zero is undefined):\[\frac{2 \sin x \cos x}{\cos x} = 2 \sin x.\]
- Correct Mistake: We realize there is a mistake in the previous step. We should have divided by \(\cos^2 x\), not \(\cos x\). Let's correct this:\(\newline\)\[(2 \sin x \cos x) / \cos^2 x = (2 \sin x) / \cos x.\]
