Sum of Squares of Sine and Cosine

Theorem

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

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

Corollaries
where:
 * $\tan$ and $\cot$ are tangent and cotangent.
 * $\sec$ and $\csc$ are secant and cosecant.

Geometric Proof

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


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


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


 * Squaring both sides and adding them together gives:


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

Unit Circle Proof
Every point $P = (x,y)$ on the unit circle is defined as $P = (\cos \theta,\sin \theta)$.

The graph of said unit circle is the locus of:


 * $x^2 + y^2 = 1$

as given by Equation of a Circle.

Substituting $x = \cos \theta$ and $y = \sin \theta$ yields:


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

Proof of Corollaries

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


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

Also see

 * Difference of Squares of Hyperbolic Cosine and Sine