QuestionJuly 15, 2025

Question 2 Prove that sinxtan(x)/(2)+2cosx=2cos^2((x)/(2))

Question 2 Prove that sinxtan(x)/(2)+2cosx=2cos^2((x)/(2))
Question 2 Prove that sinxtan(x)/(2)+2cosx=2cos^2((x)/(2))

Solution
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Answer

Proven: sinx \cdot \tan\frac{x}{2} + 2\cos x = 2\cos^{2}(\frac{x}{2}) Explanation 1. Use Trigonometric Identities Recall that \tan\frac{x}{2} = \frac{1-\cos x}{\sin x} and 2\cos^2(\frac{x}{2}) = 1 + \cos x. 2. Simplify LHS Substitute \tan\frac{x}{2} in the expression sinx \cdot \tan\frac{x}{2} + 2\cos x: \[ \sin x \cdot \frac{1-\cos x}{\sin x} + 2\cos x = (1 - \cos x) + 2\cos x \] 3. Combine Terms Simplify the expression: \[ 1 - \cos x + 2\cos x = 1 + \cos x \] 4. Compare with RHS The simplified LHS 1 + \cos x matches the RHS 2\cos^2(\frac{x}{2}).

Explanation

1. Use Trigonometric Identities<br /> Recall that $\tan\frac{x}{2} = \frac{1-\cos x}{\sin x}$ and $2\cos^2(\frac{x}{2}) = 1 + \cos x$.<br /><br />2. Simplify LHS<br /> Substitute $\tan\frac{x}{2}$ in the expression $sinx \cdot \tan\frac{x}{2} + 2\cos x$: <br />\[ \sin x \cdot \frac{1-\cos x}{\sin x} + 2\cos x = (1 - \cos x) + 2\cos x \]<br /><br />3. Combine Terms<br /> Simplify the expression:<br />\[ 1 - \cos x + 2\cos x = 1 + \cos x \]<br /><br />4. Compare with RHS<br /> The simplified LHS $1 + \cos x$ matches the RHS $2\cos^2(\frac{x}{2})$.
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