"Prove that sin⁡(x)2+sin2(x2)tan⁡(x2)→tan⁡(x2)" was answered by our experts

High School
Geometry
Trigonometry
Trigonometric equations and identities

Betsy Rhone

Answered question

2021-12-31

Prove that

\frac{\sin (x)}{2} + \sin^{2} (\frac{x}{2}) \tan (\frac{x}{2}) \to \tan (\frac{x}{2})

Answer & Explanation

Daniel Cormack

Beginner2022-01-01Added 34 answers

Use

\sin (x) = 2 \sin (\frac{x}{2}) \cos (\frac{x}{2})

then we have

\frac{\sin x}{2} + \sin^{2} (\frac{x}{2}) \tan (\frac{x}{2}) = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}} \underset{= 1}{\underset{⏟}{\cos^{2} \frac{x}{2} + \sin^{2} \frac{x}{2}}}

= \tan \frac{x}{2}

chumants6g

Beginner2022-01-02Added 33 answers

Fairly obvious:

\sin^{2} \frac{x}{2} \tan \frac{x}{2} = (1 - \cos^{2} \frac{x}{2}) \tan \frac{x}{2} = \tan \frac{x}{2} - \cos \frac{x}{2} \sin \frac{x}{2} = \tan \frac{x}{2} - \frac{\sin x}{2}

Vasquez

Expert2022-01-09Added 669 answers

With $s := \sin \frac{x}{2}, c := \cos \frac{x}{2}, \sin x = 2 s c$ and
$\frac{1}{2} 2 s c + s^{2} \frac{s}{c} = \frac{s c^{2} + s^{3}}{c} = \frac{s}{c}$

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