Prove that $ an( heta) sec( heta) = sin( heta) $ where $ heta $ is an angle in the unit circle

We start with the definitions of the trigonometric functions on the unit circle.

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$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $

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$ \sec(\theta) = \frac{1}{\cos(\theta)} $

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Multiplying these two expressions, we have:

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$ \tan(\theta) \sec(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \cdot \frac{1}{\cos(\theta)} $

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Simplifying, we get:

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$ \tan(\theta) \sec(\theta) = \frac{\sin(\theta)}{\cos^2(\theta)} \cdot \cos(\theta) $

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Since $ \cos^2(\theta) \cos(\theta) = \cos(\theta) $, we have:

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$ \tan(\theta) \sec(\theta) = \sin(\theta) \cdot \frac{1}{\cos^2(\theta)} = \sin(\theta) $

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Thus, it is proven that:

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$ \boxed{\tan(\theta) \sec(\theta) = \sin(\theta)} $

We start by writing the definitions of the trigonometric functions:

$ an( heta) = frac{sin( heta)}{cos( heta)} $

$ sec( heta) = frac{1}{cos( heta)} $

Now multiply them:

$ an( heta) sec( heta) = frac{sin( heta)}{cos( heta)} cdot frac{1}{cos( heta)} = frac{sin( heta)}{cos^2( heta)} $

Since $ cos^2( heta) = 1 $, we get:

$ an( heta) sec( heta) = sin( heta) $

We use the definitions:

$ an( heta) = frac{sin( heta)}{cos( heta)} $

$ sec( heta) = frac{1}{cos( heta)} $

Multiplying:

$ an( heta) sec( heta) = frac{sin( heta)}{cos( heta)} cdot frac{1}{cos( heta)} $

Therefore:

$ sin( heta) cdot frac{1}{cos( heta)} = sin( heta) $

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