Trigonometric Identities
Formula
The following identity holds for trigonometric functions.
$$ \begin{align} \cos^{2} x + \sin^{2} x &= 1 \\ 1 + \tan^{2} x &= \sec^{2} x \\ 1 + \cot^{2} x &= \csc^{2} x \end{align} $$
Proof
$(1)$
From the addition formula of trigonometric functions,
$$ \cos (x - y) = \cos x \cos y + \sin x \sin y $$
Substituting $y= x$,
$$ \cos 0 = \cos^{2} x + \sin^{2} x \implies \cos^{2} x + \sin^{2} x = 1 $$
■
$(2)$
Dividing both sides of $(1)$ by $\cos^{2}x$,
$$ \dfrac{cos^{2}x}{\cos^{2}x} + \dfrac{\sin^{2}x}{\cos^{2}x} = \dfrac{1}{\cos^{2}x} \implies 1 + \tan^{2} x = \sec^{2} x $$
■
$(3)$
Dividing both sides of $(1)$ by $\sin^{2}x$,
$$ \dfrac{\cos^{2}x}{\sin^{2}x} + \dfrac{\sin^{2}x}{\sin^{2}x} = \dfrac{1}{\sin^{2}x} \implies \cot^{2} x + 1 = \csc^{2} x $$
■