Inverse Trigonometric Identities

Negative Formulaes

\[ sin^{-1}(-x) = -sin^{-1}(x)\]

\[ cos^{-1}(-x) = π-cos^{-1}(x)\]

\[ tan^{-1}(-x) = -tan^{-1}(x)\]

\[ cot^{-1}(-x) = π-cot^{-1}(x)\]

\[ sec^{-1}(-x) = π-sec^{-1}(x)\]

\[ cosec^{-1}(-x) = -cosec^{-1}(x)\]

Addition of two functions

\[sin^{-1}x + cos^{-1}x = π/2 \space,\space x ∈ [-1, 1] \]

\[tan^{-1}x + cot^{-1}x = π/2 , x ∈ R\]

\[cosec^{-1}x + sec^{-1}x = π/2 \]

Reciprocal Formulaes

\[sin^{-1}\frac{1}{x} = cosec^{-1}x \]

\[cos^{-1}\frac{1}{x} = sec^{-1}x \]

\[tan^{-1}\frac{1}{x} = cot^{-1}x, x > 0\]

Identity Formulaes

\[sin(sin^{-1}x) = x, -1≤ x ≤1\]

\[cos(cos^{-1}x) = x, -1≤ x ≤1\]

\[tan(tan^{-1}x) = x, – ∞ < x < ∞ \]

\[cot(cot^{-1}x) = x, – ∞ < x < ∞ \]

\[sec(sec^{-1}x) = x, - ∞ < x ≤ 1\space or\space 1 ≤ x < ∞\]

\[cosec(cosec^{-1}x) = x, - ∞ < x ≤ 1\space or\space 1 ≤ x < ∞\]

Addition of Same functions

\[sin^{-1}x+sin^{-1}y = sin^{-1}(x\sqrt{1-y^2} + y\sqrt{1-x^2})\]

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

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

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

\[tan^{-1}x+tan^{-1}y = tan^{-1}\frac{x+y}{1-xy}\]

\[tan^{-1}x-tan^{-1}y = tan^{-1}\frac{x-y}{1+xy}\]

Simplified Inverse Trigonometric Functions

\[Equation\]	\[Conditions\]	\[Graph\]
\[y=sin^{-1}(\frac{2x}{1+x^2})\]	\[\space \space \space \space \space \space \space \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space |x| \leqslant 1\] \[ \pi \space - \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x>1\]\[-(\pi + 2tan^{-1}x) \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space \space x <-1\]
\[y=cos^{-1}(\frac{1-x^2}{1+x^2})\]	\[2tan^{-1}x \space \space \space \space \space \space \space \space \space if \space \space \space \space \space x \geqslant 0\] \[ -2tan^{-1}x \space \space \space \space \space \space if \space \space \space \space \space x <0\]
\[y=tan^{-1}(\frac{2x}{1-x^2})\]	\[\space \space \space \space \space \space \space \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space |x| <1\] \[ \pi \space + \space 2tan^{-1}x \space \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x<-1\]\[-(\pi - 2tan^{-1}x) \space \space \space \space \space \space \space \space if \space \space \space \space \space \space x>1\]
\[y=sin^{-1}(3x-4x^3)\]	\[-(\pi+3sin^{-1}x) \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x \leqslant -\frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 3sin^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space - \frac{1}{2} \leqslant x \leqslant \frac{1}{2}\]\[\pi - 3sin^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \space \frac{1}{2} \leqslant x \leqslant 1\]
\[y=cos^{-1}(4x^3-3x)\]	\[3cos^{-1}x-2\pi \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x \leqslant -\frac{1}{2}\]\[2\pi - 3cos^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space -\frac{1}{2} \leqslant x \leqslant \frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 3cos^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \frac{1}{2} \leqslant x \leqslant 1\]
\[y=sin^{-1}(2x \sqrt{1-x^2})\]	\[-(\pi+2sin^{-1}x) \space \space \space \space \space \space if \space \space \space \space \space \space -1 \leqslant x\leqslant -\frac{1}{2}\]\[\space \space \space \space \space \space \space \space \space \space \space 2sin^{-1}x \space \space \space \space \space \space \space if \space \space \space \space \space \space \space \space -\frac{1}{\sqrt{2}} \leqslant x \leqslant \frac{1}{\sqrt{2}}\]\[\pi - 2sin^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \frac{1}{\sqrt{2}} \leqslant x \leqslant 1\]
\[y=cos^{-1}(2x^2-1)\]	\[\space \space \space \space 2cos^{-1}x \space \space \space \space \space \space if \space \space \space \space \space \space \space \space \space 0 \leqslant x \leqslant 1\]\[2\pi-2cos^{-1}x \space \space \space \space \space \space if \space \space \space \space\space -1 \leqslant x \leqslant 0\]

Domain,Range and Graph of Inverse Functions

\[Function\]	\[Domain\]	\[Range \,of\, an\] \[ Inverse\, Function\]	\[Graphs\]
\[sin^{-1}x\] \[(arcsine\,x)\]	\[-1≤ x ≤1\]	\[-\frac{π}{2}≤y≤ \frac{π}{2}\]
\[cos^{-1}x\] \[(arccosine\,x)\]	\[-1≤ x ≤1\]	\[0≤y ≤π\]
\[tan^{-1}x\] \[(arctangent\,x)\]	\[– ∞ < x <  ∞\]	\[-\frac{π}{2}<y<\frac{π}{2}\]
\[cot^{-1}x\] \[(arccotangent\,x)\]	\[– ∞ < x <  ∞\]	\[0<y<π\]
\[sec^{-1}x\] \[(arcsecant\,x)\]	\[– ∞ ≤ x ≤-1 \] \[or\] \[ 1≤x≤ ∞\]	\[0≤y≤\pi,\] \[y \neq \frac{\pi}{2}\]
\[cosec^{-1}x\] \[(arccosecant\,x)\]	\[– ∞ ≤ x ≤-1\, or\, 1≤x≤ ∞\]	\[-\frac{π}{2}≤y≤\frac{π}{2},y \neq 0\]