Minor Trigonometrical Identities

Theorem

Sum of Tangent and Cotangent

$\tan x + \cot x = \sec x \csc x$

Tangent times Tangent Plus Cotangent

$\tan x \left({\tan x + \cot x}\right) = \sec^2 x$

Secant Minus Cosine

$\sec x - \cos x = \sin x \tan x$

Square of Tangent Minus Square of Sine

$\tan^2 x - \sin^2 x = \tan^2 x \ \sin^2 x$

Difference of Fourth Powers of Cosine and Sine

$\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x$

Cosecant Minus Sine

$\csc x - \sin x = \cos x \ \cot x$

Sum of Squares of Secant and Cosecant

$\sec^2 x + \csc^2 x = \sec^2 x \ \csc^2 x$

Difference of Fourth Powers of Secant and Tangent

$\sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x$

Sum of Reciprocals of One Plus and Minus Sine

$\displaystyle \frac 1 {1 - \sin x} + \frac 1 {1 + \sin x} = 2 \ \sec^2 x$

Difference of Reciprocals of One Plus and Minus Sine

$\displaystyle \frac 1 {1 - \sin x} - \frac 1 {1 + \sin x} = 2 \tan x \ \sec x$

Sum of Secant and Tangent

$\displaystyle \sec x + \tan x = \frac {1 + \sin x} {\cos x}$

Cosine over Sum of Secant and Tangent

$\displaystyle \frac {\cos x} {\sec x + \tan x} = 1 - \sin x$

Secant Plus One over Secant Squared

$\displaystyle \frac {\sec x + 1} {\sec^2 x} = \frac {\sin^2 x} {\sec x - 1}$

Sine Plus Cosine times Tangent Plus Cotangent

$\left({\sin x + \cos x}\right) \left({\tan x + \cot x}\right) = \sec x + \csc x$

Tangent over Secant Plus One

$\displaystyle \frac {\tan x} {\sec x + 1} = \frac {\sec x - 1} {\tan x}$

Squares of Linear Combination of Sine and Cosine

$\left({a \cos x + b \sin x}\right)^2 + \left({b \cos x - a \sin x}\right)^2 = a^2 + b^2$

Reciprocal of One Minus Secant

$\displaystyle \frac {\sin^2 x + 2 \cos x - 1} {\sin^2 x + 3 \cos x - 3} = \frac 1 {1 - \sec x}$

Reciprocal of One Plus Cosecant

$\displaystyle \frac {\cos^2 x + 3 \sin x - 1} {\cos^2 x + 2 \sin x + 2} = \frac 1 {1 + \csc x}$