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Differentiation

Trignometric Identities

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Trignometric Identities Index

Inverse of a term
Ratio of two terms
Pythogorean
Even/odd or negative angle
Co-Function
Two terms in angle
Double Angle
Square of a term
Sum of two terms
Product of two terms

Inverse

$\sin a = \frac{1}{\csc a}$
$\cos a = \frac{1}{\sec a}$
$\tan a = \frac{1}{\cot a}$
$\csc a = \frac{1}{\sin a}$
$\sec a = \frac{1}{\cos a}$
$\cot a = \frac{1}{\tan a}$

Ratio

$\tan a = \frac{\sin a}{\cos a}$
$\cot a = \frac{\cos a}{\sin a}$

Pythagorean

$\sin^{2} a + \cos^{2} a = 1$
$\sec^{2} a - \tan^{2} a = 1$
$\csc^{2} a - \cot^{2} a = 1$

Even/Odd

$\sin (- a) = - \sin a$
$\cos (- a) = + \cos a$
$\tan (- a) = - \tan a$
$\csc (- a) = - \csc a$
$\sec (- a) = + \sec a$
$\cot (- a) = - \cot a$

Co-Function

$\sin (\frac{π}{2} - a) = \cos a$
$\cos (\frac{π}{2} - a) = \sin a$
$\tan (\frac{π}{2} - a) = \cot a$
$\csc (\frac{π}{2} - a) = \sec a$
$\sec (\frac{π}{2} - a) = \csc a$
$\cot (\frac{π}{2} - a) = \tan a$

Two Terms in Angle

$\sin (a \pm b) = \sin a \cos b \pm \cos a \sin b$
$\cos (a \pm b) = \cos a \cos b \mp \sin a \sin b$
$\tan (a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}$

Double Angle

$\sin (2 a) = 2 \sin a \cos a, \sin a \cos a = \frac{\sin (2 a)}{2}$
$\cos (2 a) = \cos^{2} a - \sin^{2} a = 2 \cos^{2} a - 1 = 1 - 2 \sin^{2} a$
$\tan (2 a) = \frac{2 \tan a}{1 - \tan^{2} a}$

Square

$\sin^{2} a = \frac{1 - \cos (2 a)}{2}$
$\cos^{2} a = \frac{1 + \cos (2 a)}{2}$
$\tan^{2} a = \frac{1 - \cos (2 a)}{1 + \cos (2 a)}$

Sum

$\sin a + \sin b = 2 \sin (\frac{a + b}{2}) \cos (\frac{a - b}{2})$
$\sin a - \sin b = 2 \sin (\frac{a - b}{2}) \cos (\frac{a + b}{2})$
$\cos a + \cos b = 2 \cos (\frac{a + b}{2}) \cos (\frac{a - b}{2})$
$\cos a - \cos b = - 2 \sin (\frac{a + b}{2}) \sin (\frac{a - b}{2})$

Product

$\sin a \sin b = \frac{1}{2} [\cos (a - b) - \cos (a + b)]$
$\cos a \cos b = \frac{1}{2} [\cos (a - b) + \cos (a + b)]$
$\sin a \cos b = \frac{1}{2} [\sin (a + b) + \sin (a - b)]$
$\cos a \sin b = \frac{1}{2} [\sin (a + b) - \sin (a - b)]$

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