📂Functions

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 $y= x$,

$$\cos 0 = \cos^{2} x + \sin^{2} x \implies \cos^{2} x + \sin^{2} x = 1$$

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$(2)$

Dividing both sides of $(1)$ by $\cos^{2}x$,

$$\dfrac{cos^{2}x}{\cos^{2}x} + \dfrac{\sin^{2}x}{\cos^{2}x} = \dfrac{1}{\cos^{2}x} \implies 1 + \tan^{2} x = \sec^{2} x$$

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$(3)$

Dividing both sides of $(1)$ by $\sin^{2}x$,

$$\dfrac{\cos^{2}x}{\sin^{2}x} + \dfrac{\sin^{2}x}{\sin^{2}x} = \dfrac{1}{\sin^{2}x} \implies \cot^{2} x + 1 = \csc^{2} x$$

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