(i) let \(\sin^{-1}\left(\frac{-1}{\sqrt{2}}\right)=\mathrm x\)

⇒\(-\sin^{-1}\left(\frac{1}{\sqrt{2}}\right)=\mathrm x\) [Formula: sin ^-1(-x) = -sin ^-1 x ]

⇒ \(\frac{1}{\sqrt{2}}=-\sin\mathrm x\) 

[We know which value of x when put in this expression will give us this result]

\(\therefore \mathrm x=-\frac{\pi}{4}\)

(ii) \(cos^{-1}\left(\frac{-\sqrt{3}}{2}\right)\)\(=\pi-cos^{-1}\left(\frac{\sqrt{3}}{2}\right)\) 

[ Formula: cos -1(-x) = π – cos -1 x]

let \(cos^{-1}\left(\frac{-\sqrt{3}}{2}\right)=\mathrm x\)

⇒\(\left(\frac{\sqrt{3}}{2}\right)=\cos\mathrm x\)

[We know which value of x when put in this expression will give us this result]

\(\therefore \mathrm x=\frac{\pi}{6}\)

Putting this value back in the equation

\(\pi-\frac{\pi}{6}=\frac{5\pi}{6}\)

(iii) let \(\tan^{-1}(-\sqrt{3})=\mathrm x\)

⇒ \(-\tan^{-1}(\sqrt{3})=\mathrm x\)  [Formula: tan ^-1(-x) = - tan ^-1 (x)]

⇒\(\sqrt{3}=-\tan\mathrm x\)

[We know which value of x when put in this expression will give us this result]

\(\therefore \mathrm x=\frac{-\pi}{3}\)

(iv) \(\sec^{-1}(-2)=\pi-\sec^{-1}(2)\) ......(i)

[ Formula: sec ^-1(-x) = π– sec ^-1 (x) ]

let \(\sec^{-1}(-2)=\mathrm x\)

⇒ 2 = sec x 

[We know which value of x when put in this expression will give us this result]

\(\therefore \mathrm x=\frac{\pi}{3}\)

Putting the value in (i)

\(\pi-\frac{\pi}{3}=\frac{2\pi}{3}\)

(v) let \(cosec^{-1}(-\sqrt{2})=\mathrm x\)

⇒ \(-cosec^{-1}(\sqrt{2})=\mathrm x\)

[ Formula: cosec ^-1(-x) = -cosec ^-1 (x) ]

⇒ \(\sqrt{2}=-cosec\mathrm x\)

\(\therefore \mathrm x=-\frac{\pi}{4}\)

(vi) \(\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)\)\(=\pi-\cot^{-1}\left(\frac{1}{\sqrt{3}}\right)\) .....(i)

let \(\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)\) = x

⇒ \(\frac{1}{\sqrt{3}}=\cot^{-1}\mathrm x\)

[We know which value of x when put in this expression will give us this result]

⇒\(\mathrm x=\frac{\pi}{3}\)

Putting in (i)

\(\pi-\frac{\pi}{3}=\frac{2\pi}{3}\)