Find the principal value of :

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(i) Let

⇒ [Formula: sin^{-1}(-x) = -sin^{-1} x ]

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

∴

(ii) [ Formula: cos^{-1}(-x) = π – cos^{-1} x]

Let

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

∴

Putting this value back in the equation

(iii) Let

⇒ [Formula: tan^{-1}(-x) = - tan^{-1} (x)]

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

∴

(iv) …(i) [ Formula:sec^{-1}(-x) = π– sec^{-1} (x) ]

Let

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

∴

Putting the value in (i)

(v) Let

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

⇒

∴

(vi) … (i)

Let

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

⇒

Putting in (i)

=

1