All Inverse trigonometry functions and formulas or Identity list | pdf

 Inverse trigonometry functions and formulas

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Negative Property

sin^-1 (-x) = -sin^-1 x

cos^-1 (-x) = π - cos^-1 x

tan^-1 (-x) = -tan^-1 x

cot^-1 (-x) = π - cot^-1 x

sec^-1 (-x) = π - sec^-1 x

cosec^-1 (-x) = -cosec^-1 x

Complementary Property

sin^-1 x + cos^-1 x = \(\frac {π}{2}\)

tan^-1 x + cot^-1 x = \(\frac {π}{2}\)

sec^-1 x + cosec^-1 x = \(\frac {π}{2}\)

Properties of Inverse Circular function

sin^-1 x = cosec^-1 \(\frac {1}{x}\)

cosec^-1 x = sin^-1 \(\frac {1}{x}\)

cos^-1 x = sec^-1 \(\frac {1}{x}\)

sec^-1 x = cos^-1 \(\frac {1}{x}\)

tan^-1 x = cot^-1 \(\frac {1}{x}\)

cot^-1 x = tan^-1 \(\frac {1}{x}\)

Self Adjusting Property

sin^-1 (sin x) = x   :   x ∈ \(\left [ \frac {-π}{2}, \frac {π}{2} \right ] \)

cos^-1 (cos x) = x   :   x ∈ \(\left [ 0, π \right ] \)

tan^-1 (tan x) = x   :   x ∈ \(\left ( \frac {-π}{2}, \frac {π}{2} \right ) \)

cot^-1 (cot x) = x   :   x ∈ \( \left ( 0, π \right ) \)

sec^-1 (sec x) = x   :   x ∈ [0, π] - \(  \left\{ \frac {π}{2} \right\} \)

cosec^-1 (cosec x) = x   :   x ∈ \(\left [ \frac {-π}{2}, \frac {π}{2} \right ] - \left\{0\right\} \)

sin (sin^-1 x) = x   :   x ∈ \(\left [ -1, 1 \right ] \)

cos (cos^-1 x) = x   :   x ∈ \(\left [ -1, 1 \right ] \)

tan (tan^-1 x) = x   :   x ∈ R

cot (cot^-1 x) = x   :   x ∈ R

sec (sec^-1 x) = x   :   x ∈ R - (-1, 1)

cosec (cosec^-1 x) = x   :   x ∈ R - (-1, 1)

Addition and Subtraction formulas

sin^-1 x + sin^-1 y = sin^-1 \(x\sqrt {1 - y^2} + y\sqrt {1 - x^2}\);  x ≥ 0, y ≥ 0 & x² + y² ≤ 1

sin^-1 x + sin^-1 y = π - sin^-1 \(x\sqrt {1 - y^2} + y\sqrt {1 - x^2}\);  x ≥ 0, y ≥ 0 & x² + y² ≥ 1

sin^-1 x - sin^-1 y = sin^-1 \(x\sqrt {1 - y^2} - y\sqrt {1 - x^2}\);  x,y ∈ [0, 1]

cos^-1 x + cos^-1 y = cos^-1 \(xy - \sqrt {1 - x^2}\sqrt {1 - y^2}\);  x,y ∈ [0, 1]

cos^-1 x - cos^-1 y = cos^-1 \(xy + \sqrt {1 - x^2}\sqrt {1 - y^2}\);  0 ≤ x < y ≤ 1

cos^-1 x - cos^-1 y = - cos^-1 \(xy + \sqrt {1 - x^2}\sqrt {1 - y^2}\);  0 ≤ y < x ≤ 1

tan^-1 x + tan^-1 y = tan^-1 \(\frac{x+y}{1 - xy}\);  x,y > 0 & xy < 1

tan^-1 x + tan^-1 y = π + tan^-1 \(\frac{x+y}{1 - xy}\);  x,y > 0 & xy > 1

tan^-1 x + tan^-1 y = π/2 if x,y > 0 & xy = 1

tan^-1 x + tan^-1 y = -π/2 if x,y < 0 & xy = 1

tan^-1 x - tan^-1 y = tan^-1 \(\frac{x-y}{1 + xy}\);  x,y ≥ 0 & xy > -1

2tan^-1 x = sin^-1 \(\frac {2x}{1+x^2}\);  -1 ≤ x ≤ 1

2tan^-1 x = tan^-1 \(\frac {2x}{1-x^2}\);   -1 < x < 1

2tan^-1 x = cos^-1 \(\frac {1-x^2}{1+x^2}\);   x ≥ 0

2sin^-1 x = sin^-1 \(2x\sqrt {1 - x^2}\)

3sin^-1 x = sin^-1 (3x - 4x³ )

2cos^-1 x = cos^-1 (2x² - 1)

3cos^-1 x = cos^-1 (4x³ - 3x)

3tan^-1 x = tan^-1 \( \left ( \frac {3x - x^3}{1 - 3x^2} \right ) \)

Inverse Trigonometric Functions Series or Progressions

Taylor's Expansions

\(sin^{-1}x = x + \frac{x^3}{3!} + \frac{9x^5}{5!} + \frac{45x^7}{7!}+ ......\)

\(cos^{-1}x = \frac{π}{2}- \left( x + \frac{x^3}{3!} + \frac{9x^5}{5!} + \frac{45x^7}{7!}+ ......\right) \)

\(tan^{-1}x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7}+ ......\)

\(sec^{-1}x = cos^{-1}\frac{1}{x}\)

   \( = \frac{π}{2}- \left( \frac{1}{x} + \frac{1}{x^33!} + \frac{9}{x^55!} + \frac{45}{x^77!}+ ......\right) \)

\(cosec^{-1}x = \frac{1}{x} + \frac{1}{x^33!} + \frac{9}{x^55!} + \frac{45}{x^77!}+ ......\)

\(cot^{-1}x = \frac{π}{2}- \left( x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7}+ ......\right)\)

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