Minor Trigonometric Identities
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Theorem
Sum of Tangent and Cotangent
- $\tan x + \cot x = \sec x \csc x$
Tangent times Tangent plus Cotangent
- $\tan x \paren {\tan x + \cot x} = \sec^2 x$
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
- $\csc x - \sin x = \cos x \ \cot x$
Cotangent Minus Tangent
- $\cot x - \tan x = 2 \cot 2 x$
Sum of Cosecant and Cotangent
- $\csc x + \cot x = \cot {\dfrac x 2}$
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
- $\sec^4 x - \tan^4 x = \sec^2 x + \tan^2 x$
Reciprocal of One Plus Sine
- $\dfrac 1 {1 + \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 - \dfrac x 2}$
Reciprocal of One Minus Sine
- $\dfrac 1 {1 - \sin x} = \dfrac 1 2 \map {\sec^2} {\dfrac \pi 4 + \dfrac x 2}$
Sum of Reciprocals of One Plus and Minus Sine
- $\dfrac 1 {1 - \sin x} + \dfrac 1 {1 + \sin x} = 2 \sec^2 x$
Difference of Reciprocals of One Plus and Minus Sine
- $\ds \frac 1 {1 - \sin x} - \frac 1 {1 + \sin x} = 2 \tan x \sec x$
Reciprocal of One Plus Cosine
- $\dfrac 1 {1 + \cos x} = \dfrac 1 2 \sec^2 \dfrac x 2$
Reciprocal of One Minus Cosine
- $\dfrac 1 {1 - \cos x} = \dfrac 1 2 \map {\csc^2} {\dfrac x 2}$
Sum of Secant and Tangent
- $\sec x + \tan x = \dfrac {1 + \sin x} {\cos x}$
Cosine over Sum of Secant and Tangent
- $\dfrac {\cos x} {\sec x + \tan x} = 1 - \sin x$
Secant Plus One over Secant Squared
- $\dfrac {\sec x + 1} {\sec^2 x} = \dfrac {\sin^2 x} {\sec x - 1}$
Sine Plus Cosine times Tangent Plus Cotangent
- $\paren {\sin x + \cos x} \paren {\tan x + \cot x} = \sec x + \csc x$
Tangent over Secant Plus One
- $\dfrac {\tan x} {\sec x + 1} = \dfrac {\sec x - 1} {\tan x}$
Squares of Linear Combination of Sine and Cosine
- $\paren {a \cos x + b \sin x}^2 + \paren {b \cos x - a \sin x}^2 = a^2 + b^2$
Reciprocal of One Minus Secant
- $\dfrac {\sin^2 x + 2 \cos x - 1} {\sin^2 x + 3 \cos x - 3} = \dfrac 1 {1 - \sec x}$
Reciprocal of One Plus Cosecant
- $\dfrac {\cos^2 x + 3 \sin x - 1} {\cos^2 x + 2 \sin x + 2} = \dfrac 1 {1 + \csc x}$