Where Does the Tangent Half-Angle Identity Come From?

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In summary, the Tangent Half-Angle Identity is a trigonometric identity used to relate the tangent of a half-angle to the tangent of the original angle. The formula for this identity is tan(θ/2) = ±√[(1-cosθ)/(1+cosθ)], and it is derived using the double angle formula for tangent and the Pythagorean identity for sine and cosine. The purpose of this identity is to simplify trigonometric expressions and solve equations involving half-angles. The ± symbol in the formula indicates that there are two possible solutions, allowing for use in all four quadrants.
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Yuqing
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I saw this identity somewhere and have been looking for a derivation but I can't seem to find one. It would be of great help if someone can show me where this comes from.

[tex]tan\frac{\theta}{2} = csc\theta - cot\theta [/tex]
 
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  • #2
Hi Yuqing! :smile:

(have a theta: θ :wink:)

Hint: cscθ - cotθ = (1 - cosθ)/sinθ = … ? :smile:
 
  • #3
Ah I see. Didn't think it was so simple.

Thank you very much.
 

FAQ: Where Does the Tangent Half-Angle Identity Come From?

What is the Tangent Half-Angle Identity?

The Tangent Half-Angle Identity is a trigonometric identity that relates the tangent of a half-angle to the tangent of the original angle.

What is the formula for the Tangent Half-Angle Identity?

The formula for the Tangent Half-Angle Identity is tan(θ/2) = ±√[(1-cosθ)/(1+cosθ)].

What is the purpose of the Tangent Half-Angle Identity?

The Tangent Half-Angle Identity is used to simplify trigonometric expressions and solve trigonometric equations involving half-angles.

How is the Tangent Half-Angle Identity derived?

The Tangent Half-Angle Identity can be derived using the double angle formula for tangent and the Pythagorean identity for sine and cosine.

What is the significance of the ± symbol in the Tangent Half-Angle Identity formula?

The ± symbol indicates that there are two possible solutions for the tangent of the half-angle, depending on the quadrant of the original angle. This allows for the use of the Tangent Half-Angle Identity in all four quadrants.

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