Sine of Angle plus Right Angle

Theorem

$\sin \paren {x + \dfrac \pi 2} = \cos x$

Proof

 $\displaystyle \sin \paren {x + \frac \pi 2}$ $=$ $\displaystyle \sin x \cos \frac \pi 2 + \cos x \sin \frac \pi 2$ Sine of Sum $\displaystyle$ $=$ $\displaystyle \sin x \cdot 0 + \cos x \cdot 1$ Cosine of Right Angle and Sine of Right Angle $\displaystyle$ $=$ $\displaystyle \cos x$

$\blacksquare$