\(\displaystyle 2).\;\; \sin ^{-1}(x)+\cos ^{-1}(x)=\frac{\pi}{2}\)
\(\displaystyle 3).\;\; \tan ^{-1}(-x)=-\tan ^{-1}(x)\)
\(\displaystyle 4).\;\; \sin ^{-1}(-x)=-\sin ^{-1}(x), \quad x \in[-1,1]\)
\(\displaystyle 5).\;\; \cos ^{-1}(-x)=\pi-\cos ^{-1}(x), \quad x \in[-1,1]\)
\(\displaystyle 6).\;\; \tan ^{-1}(-x)=-\tan ^{-1}(x), \quad x \in \mathbb{R}\)
\(\displaystyle 7).\;\; \operatorname{cosec}^{-1}(-x)=-\operatorname{cosec}^{-1}(x), \quad|x| \geq 1\)
\(\displaystyle 8).\;\; \sec ^{-1}(-x)=\pi-\sec ^{-1}(x), \quad|x| \geq 1\)
\(\displaystyle 9).\;\; \cot ^{-1}(-x)=\pi-\cot ^{-1}(x), \quad x \in R\)
\(\displaystyle 10).\;\; \sin ^{-1} x+\cos ^{-1} x=\frac{\pi}{2}, \quad x \in[-1,1] \)
\(\displaystyle 11).\;\; \tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2}, \quad x \in R\)
\(\displaystyle 12).\;\; \sec ^{-1} x+\operatorname{cosec}^{-1} x=\frac{\pi}{2} \quad |x| \geq 1\)
\(\displaystyle 13).\;\; \sin ^{-1}\left(\frac{1}{x}\right)=\operatorname{cosec}^{-1}x ,\;\; if \; x \geq 1 \; or \;x \leq-1\)
\(\displaystyle 14).\;\; \cos ^{-1}\left(\frac{1}{x}\right)=\sec^{-1}x \;\;if \;\; x;\; \geq 1 \;\;or\;\; x \leq-1\)
\(\displaystyle 15).\;\; \tan ^{-1}\left(\frac{1}{x}\right)=\cot ^{-1}(x), x>0\)
\(\displaystyle 16).\;\; \tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)\) , if the value x y < 1
\(\displaystyle 17).\;\; \tan ^{-1} x-\tan ^{-1} y=\tan ^{-1}\left(\frac{x-y}{1+x y}\right) \) , if the value x y>-1
\(\displaystyle 18).\;\; 2 \tan ^{-1} x=\sin ^{-1}\left(\frac{2 x}{\left(1+x^{2}\right)}\right) \;\; |x| \geq 1\)
\(\displaystyle 19).\;\; 2 \tan ^{-1} x=\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) \quad x \geq 0 \)
\(\displaystyle 20).\;\; 2 \tan ^{-1} x=\tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right)\;\;\)-1 < x < 1
\(\displaystyle 21).\;\; 3 \sin ^{-1} x=\sin ^{-1}\left(3 x-4 x^{3}\right)\)
\(\displaystyle 22).\;\; 3 \cos ^{-1} x=\cos ^{-1}\left(4 x^{3}-3 x\right) \)
\(\displaystyle 23). \;\; 3 \tan ^{-1} x=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right) \)
\(\displaystyle 24).\;\; \sin \left(\sin ^{-1} x\right)=x \;\;-1 \leq x \leq 1 \)
\(\displaystyle 25).\;\; \left.\cos \left(\cos ^{-1} x\right)\right)=x \;\;\;\;-1 \leq x \leq 1 \)
\(\displaystyle 26).\;\; \left.\tan \left(\tan ^{-1} x\right)\right)=x \;\;\;\;-\infty< x < \infty \)
\(\displaystyle 27) \;\; \operatorname{cosec} \left({\operatorname{cosec}}^{-1} x \right)=x , \;\;\;\;-\infty < x \leq 1 \;or\; -1 \leq x < \infty \)
\(\displaystyle 28).\;\; \sec \left(\sec ^{-1} x\right)=x \;\;\;\;-\infty < x \leq 1 \;or\; 1 \leq x < \infty \)
\(\displaystyle 29).\;\; \cot \left(\cot ^{-1} x\right)=x \)
\(\displaystyle 29).\;\; \cot \left(\cot ^{-1} x\right)=x \;\;\;\; \pi < x < \infty\)
\(\displaystyle 30).\;\; \sin ^{-1}(\sin \theta)=\theta \;\;-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \)
\(\displaystyle 31).\;\; \cos ^{-1}(\cos \theta)=\theta \;\; 0 \leq \theta \leq \pi \)
\(\displaystyle 32).\;\; \tan ^{-1}(\tan \theta)=\theta \;\;-\frac{\pi}{2} < \theta < \frac{\pi}{2} \)
\(\displaystyle 33).\;\; \sec ^{-1}(\sec \theta)=\theta, \;\;\;\; 0 \leq \theta \leq \frac{\pi}{2} \;or\; \frac{\pi}{2}<\theta \leq \pi \)
\(\displaystyle 34).\;\; \cot ^{-1}(\cot \theta)=\theta,\;\;\;\; 0<\theta<\pi \)
\(\displaystyle 36). \;\; \sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right) ,\;\text{if}\; x, y \geq 0 \; \text{and}\; x^{2}+y^{2} \leq 1 \)
\(\displaystyle 36). \;\; \sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right) ,\;\text{if}\; x, y \geq 0 \; \text{and}\; x^{2}+y^{2} \leq 1 \)
\(\displaystyle 37).\;\; \sin ^{-1} x+\sin ^{-1} y=\pi-\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right) \;\; if\; x, y \geq 0 \; \text{and}\; x^{2}+y^{2}>1\)
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