inverse trignometry formula detailed

(i)  sin (sin−1${}^{-1}$ x) = x and sin−1${}^{-1}$ (sin θ) = θ, provided that - π2$\frac{\pi }{2}$ ≤ θ ≤ π2$\frac{\pi }{2}$ and - 1 ≤ x ≤ 1.

(ii) cos (cos−1${}^{-1}$ x) = x and cos−1${}^{-1}$ (cos θ) = θ, provided that 0 ≤ θ ≤ π and - 1 ≤ x ≤ 1.

(iii) tan (tan−1${}^{-1}$ x) = x and tan−1${}^{-1}$ (tan θ) = θ, provided that - π2$\frac{\pi }{2}$ < θ < π2$\frac{\pi }{2}$ and - ∞ < x < ∞.

(iv) csc (csc−1${}^{-1}$ x) = x and sec−1${}^{-1}$ (sec θ) = θ, provided that - π2$\frac{\pi }{2}$ ≤ θ < 0 or  0 < θ ≤ π2$\frac{\pi }{2}$  and - ∞ < x ≤ 1 or -1 ≤ x < ∞.

(v) sec (sec−1${}^{-1}$ x) = x and sec−1${}^{-1}$ (sec θ) = θ, provided that 0 ≤ θ ≤ π2$\frac{\pi }{2}$ or π2$\frac{\pi }{2}$ <  θ ≤ π and - ∞ < x ≤ 1 or 1 ≤ x < ∞.

(vi)  cot (cot−1${}^{-1}$ x) = x and cot−1${}^{-1}$ (cot θ) = θ, provided that 0 < θ < π and - ∞ < x < ∞.

(vii) The function sin−1${}^{-1}$ x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of sin−1${}^{-1}$ x then - π2$\frac{\pi }{2}$ ≤ θ ≤ π2$\frac{\pi }{2}$.

(viii) The function cos−1${}^{-1}$  x is defined if – 1 ≤ x ≤ 1; if θ be the principal value of cos−1${}^{-1}$ x then 0 ≤ θ ≤ π.

(ix) The function tan−1${}^{-1}$ x is defined for any real value of x i.e., - ∞ < x < ∞; if θ be the principal value of tan−1${}^{-1}$ x then - π2$\frac{\pi }{2}$ < θ < π2$\frac{\pi }{2}$.

(x)  The function cot−1${}^{-1}$ x is defined when - ∞ < x < ∞; if θ be the principal value of cot−1${}^{-1}$ x then - π2$\frac{\pi }{2}$ < θ < π2$\frac{\pi }{2}$ and θ ≠ 0.

(xi) The function sec−1${}^{-1}$ x is defined when, I x I ≥ 1 ; if θ be the principal value of sec−1${}^{-1}$ x then 0 ≤ θ ≤ π and θ ≠ π2$\frac{\pi }{2}$.

(xii) The function csc−1${}^{-1}$ x is defined if I x I ≥ 1; if θ be the principal value of csc−1${}^{-1}$ x then - π2$\frac{\pi }{2}$ < θ < π2$\frac{\pi }{2}$ and θ ≠ 0.

(xiii) sin−1${}^{-1}$ (-x) = - sin−1${}^{-1}$ x

(xiv) cos−1${}^{-1}$ (-x) = π - cos−1${}^{-1}$ x

(xv) tan−1${}^{-1}$ (-x) = - tan−1${}^{-1}$ x

(xvi) csc−1${}^{-1}$ (-x) = - csc−1${}^{-1}$ x

(xvii) sec−1${}^{-1}$ (-x) = π - sec−1${}^{-1}$ x

(xviii) cot−1${}^{-1}$ (-x) = cot−1${}^{-1}$ x

(xix) In numerical problems principal values of inverse circular functions are generally taken.  

(xx) sin−1${}^{-1}$ x + cos−1${}^{-1}$ x = π2$\frac{\pi }{2}$

(xxi) sec−1${}^{-1}$ x + csc−1${}^{-1}$ x = π2$\frac{\pi }{2}$.

(xxii) tan−1${}^{-1}$ x + cot−1${}^{-1}$ x = π2$\frac{\pi }{2}$

(xxiii) sin−1${}^{-1}$ x + sin−1${}^{-1}$ y = sin−1${}^{-1}$ (x 1−y2−−−−−√$\sqrt{1-y^2}$ + y1−x2−−−−−√$\sqrt{1-x^2}$), if x, y ≥ 0 and x2${}^2$  + y2${}^2$ ≤ 1.

(xxiv) sin−1${}^{-1}$ x + sin−1${}^{-1}$ y = π - sin−1${}^{-1}$ (x 1−y2−−−−−√$\sqrt{1-y^2}$ + y1−x2−−−−−√$\sqrt{1-x^2}$), if x, y ≥ 0 and x2${}^2$  + y2${}^2$ > 1.

(xxv) sin−1${}^{-1}$ x - sin−1${}^{-1}$ y = sin−1${}^{-1}$ (x 1−y2−−−−−√$\sqrt{1-y^2}$ - y1−x2−−−−−√$\sqrt{1-x^2}$), if x, y ≥ 0 and x2${}^2$  + y2${}^2$ ≤ 1.

(xxvi) sin−1${}^{-1}$ x - sin−1${}^{-1}$ y = π - sin−1${}^{-1}$ (x 1−y2−−−−−√$\sqrt{1-y^2}$ - y1−x2−−−−−√$\sqrt{1-x^2}$), if x, y ≥ 0 and x2${}^2$  + y2${}^2$ > 1.

(xxvii) cos−1${}^{-1}$ x + cos−1${}^{-1}$ y = cos−1${}^{-1}$(xy - 1−x2−−−−−√$\sqrt{1-x^2}$1−y2−−−−−√$\sqrt{1-y^2}$), if x, y > 0 and x2${}^2$  + y2${}^2$ ≤  1.

(xxviii) cos−1${}^{-1}$ x + cos−1${}^{-1}$ y = π - cos−1${}^{-1}$(xy - 1−x2−−−−−√$\sqrt{1-x^2}$1−y2−−−−−√$\sqrt{1-y^2}$), if x, y > 0 and x2${}^2$  + y2${}^2$ >  1.

(xxix) cos−1${}^{-1}$ x - cos−1${}^{-1}$ y = cos−1${}^{-1}$(xy + 1−x2−−−−−√$\sqrt{1-x^2}$1−y2−−−−−√$\sqrt{1-y^2}$), if x, y > 0 and x2${}^2$  + y2${}^2$ ≤  1.

(xxx) cos−1${}^{-1}$ x - cos−1${}^{-1}$ y = π - cos−1${}^{-1}$(xy + 1−x2−−−−−√$\sqrt{1-x^2}$1−y2−−−−−√$\sqrt{1-y^2}$), if x, y > 0 and x2${}^2$  + y2${}^2$ >  1.

(xxxi) tan−1${}^{-1}$ x + tan−1${}^{-1}$ y = tan−1${}^{-1}$ (x+y1−xy$\frac{x+y}{1-xy}$), if x > 0, y > 0 and xy < 1.

 (xxxii) tan−1${}^{-1}$ x + tan−1${}^{-1}$ y = π + tan−1${}^{-1}$ (x+y1−xy$\frac{x+y}{1-xy}$), if x > 0, y > 0 and xy > 1.

(xxxiii) tan−1${}^{-1}$ x + tan−1${}^{-1}$ y = tan−1${}^{-1}$ (x+y1−xy$\frac{x+y}{1-xy}$) - π, if x < 0, y > 0 and xy > 1.

(xxxiv) tan−1${}^{-1}$ x + tan−1${}^{-1}$ y + tan−1${}^{-1}$ z = tan−1${}^{-1}$ x+y+z−xyz1−xy−yz−zx$\frac{x+y+z-xyz}{1-xy-yz-zx}$

(xxxv) tan−1${}^{-1}$ x - tan−1${}^{-1}$ y = tan−1${}^{-1}$ (x−y1+xy$\frac{x-y}{1+xy}$)

(xxxvi) 2 sin−1${}^{-1}$ x = sin−1${}^{-1}$ (2x1−x2−−−−−√$\sqrt{1-x^2}$)

(xxxvii) 2 cos−1${}^{-1}$ x = cos−1${}^{-1}$ (2x2${}^2$ - 1)

(xxxviii) 2 tan−1${}^{-1}$ x = tan−1${}^{-1}$ (2x1−x2$\frac{2x}{1-x^2}$) = sin−1${}^{-1}$ (2x1+x2$\frac{2x}{1+x^2}$) = cos−1${}^{-1}$ (1−x21+x2$\frac{1-x^2}{1+x^2}$)

(xxxix) 3 sin−1${}^{-1}$ x = sin−1${}^{-1}$ (3x - 4x3${}^3$)

(xxxx) 3 cos−1${}^{-1}$ x = cos−1${}^{-1}$ (4x3${}^3$ - 3x)

(xxxxi) 3 tan−1${}^{-1}$ x = tan−1${}^{-1}$ (3x−x31−3x2$\frac{3x-x^3}{1-3x^2}$)

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