Sum of Squares of Sine and Cosine

Theorem

 * $$\cos^2 x + \sin^2 x = 1$$

where $$\sin$$ and $$\cos$$ are sine and cosine.

Corollaries

 * $$1 + \tan^2 x = \sec^2 x$$ (when $$\cos x \ne 0$$)


 * $$1 + \cot^2 x = \csc^2 x$$(when $$\sin x \ne 0$$)

Algebraic Proof
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Geometric Proof

 * Starting with $$\sin x$$ and $$\cos x$$,

$$\sin x = \frac{\text{opposite}}{\text{hypotenuse}}$$

$$\cos x = \frac{\text{adjacent}}{\text{hypotenuse}}$$


 * Squaring both sides and adding them together gives:

$$\sin^2 x + \cos^2 x = \frac{\text{opposite}^2 + \text{adjacent}^2}{\text{hypotenuse}^2} = 1$$ by Pythagoras's Theorem

Proof of Corollaries

 * $$1 + \tan^2 x = \sec^2 x$$ (when $$\cos x \ne 0$$):

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 * $$1 + \cot^2 x = \csc^2 x$$(when $$\sin x \ne 0$$):

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