Resolved: "How can I prove the identity 11+sin⁡(x)≡sec2(x2)(tan⁡(x2)+1)2"

Pamela Meyer

Answered question

2022-01-03

How can I prove the identity

\frac{1}{1 + \sin (x)} \equiv \frac{\sec^{2} (\frac{x}{2})}{{(\tan (\frac{x}{2}) + 1)}^{2}}

Marcus Herman

Beginner2022-01-04Added 41 answers

Use

\frac{\sec \frac{x}{2}}{\tan \frac{x}{2} + 1} \equiv \frac{1}{\sin \frac{x}{2} + \cos \frac{x}{2}}

Now you have already identified

{(\sin \frac{x}{2} + \cos \frac{x}{2})}^{2} = ?

stomachdm

Beginner2022-01-05Added 33 answers

We can express any value trigonometric ratio of a given angle

2 θ

in terms of the tangent of half that angle. For example:

\sin 2 θ \equiv \frac{2 \tan θ}{1 + \tan^{2} θ}

\cos 2 θ \equiv \frac{1 - \tan^{2} θ}{1 + \tan^{2} θ}

\tan 2 θ \equiv \frac{2 \tan θ}{1 - \tan^{2} θ}

As an aside, this is useful in integration. So, you may want to use the first identity given to prove your identity.

Vasquez

Expert2022-01-09Added 669 answers

Alternately, use half angle theorems:
$\sin x \equiv \frac{2 \tan (\frac{x}{2})}{1 + \tan^{2} (\frac{x}{2})}$

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