# How do you prove #tan^2(1/2theta)=(tantheta-sintheta)/(tantheta+sintheta)#?

see below

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To prove the identity ( \tan^2\left(\frac{1}{2}\theta\right) = \frac{\tan\theta - \sin\theta}{\tan\theta + \sin\theta} ), we'll use trigonometric identities and properties. The key identities we'll use involve the half-angle and double-angle formulas for tangent and sine, as well as the fundamental identity ( \sin^2\theta + \cos^2\theta = 1 ).

**Step 1: Start with the right side of the identity.**

Given: [ \frac{\tan\theta - \sin\theta}{\tan\theta + \sin\theta} ]

**Step 2: Use the definition of ( \tan\theta = \frac{\sin\theta}{\cos\theta} ).**

This transforms the expression into: [ \frac{\frac{\sin\theta}{\cos\theta} - \sin\theta}{\frac{\sin\theta}{\cos\theta} + \sin\theta} ]

To simplify, get a common denominator in both the numerator and the denominator: [ \frac{\frac{\sin\theta - \sin\theta\cos\theta}{\cos\theta}}{\frac{\sin\theta + \sin\theta\cos\theta}{\cos\theta}} ]

Simplifying the fractions by multiplying both numerator and denominator by (\cos\theta), we get: [ \frac{\sin\theta(1 - \cos\theta)}{\sin\theta(1 + \cos\theta)} ]

Since ( \sin\theta ) is in both the numerator and the denominator, it can be canceled out (assuming ( \theta ) is not at an angle where ( \sin\theta = 0 )), leading to: [ \frac{1 - \cos\theta}{1 + \cos\theta} ]

**Step 3: Apply the half-angle identity for tangent.**

The half-angle identity for ( \tan ) is given by: [ \tan^2\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{1 + \cos\theta} ]

This matches our derived expression exactly.

Thus, we have shown that: [ \tan^2\left(\frac{1}{2}\theta\right) = \frac{\tan\theta - \sin\theta}{\tan\theta + \sin\theta} ]

This concludes the proof. Note that this identity and the steps taken assume ( \theta ) is within a range that prevents division by zero and undefined expressions.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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