identities

Summary of Trigonometric Identities

Basic Identities
Click here for the derivation of basic identities.

1. $\sin \theta = \dfrac{1}{\csc \theta}; \,\, \csc \theta = \dfrac{1}{\sin \theta}$
2. $\cos \theta = \dfrac{1}{\sec \theta}; \,\, \sec \theta = \dfrac{1}{\cos \theta}$
3. $\tan \theta = \dfrac{\sin \theta}{\cos \theta} = \dfrac{1}{\cot \theta}$
4. $\cot \theta = \dfrac{\cos \theta}{\sin \theta} = \dfrac{1}{\tan \theta}$
 

Derivation of Pythagorean Identities

Right triangle with sides a, b, and c and angle thetaIn reference to the right triangle shown and from the functions of a right triangle:
a/c = sin θ
b/c = cos θ
c/b = sec θ
c/a = csc θ
a/b = tan θ
b/a = cot θ
 

Derivation of Sum and Difference of Two Angles

Triangle used in sum and difference of two anglesThe sum and difference of two angles can be derived from the figure shown below.
 

Derivation of the Double Angle Formulas

The Double Angle Formulas can be derived from Sum of Two Angles listed below:
$\sin (A + B) = \sin A \, \cos B + \cos A \, \sin B$   →   Equation (1)

$\cos (A + B) = \cos A \, \cos B - \sin A \, \sin B$   →   Equation (2)

$\tan (A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \, \tan B}$   →   Equation (3)
 

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