If \({\rm{co}}{{\rm{s}}^{ - 1}}\left( {\frac{1}{{\sqrt 5 }}} \right) = {\rm{\theta }}\), then what is the value of \({\rm{cose}}{{\rm{c}}^{ - 1}}\left( {\sqrt 5 } \right)?\)

Question

Question

Answer 1

Concept:

sec^-1 x = cos^-1 (1/x)

cosec^-1 (x) + sec^-1 (x) = π / 2

Calculation:

As we know that,

sec-1 x = cos-1 (1/x)

\({\rm{co}}{{\rm{s}}^{ - 1}}\left( {\frac{1}{{\sqrt 5 }}} \right) = {\rm{\theta }} \)

\(\Rightarrow {\rm{se}}{{\rm{c}}^{ - 1}}\left( {\sqrt 5 } \right) = {\rm{\theta }}\)

As we know that,

cosec-1 (x) + sec-1 (x) = π / 2

\(\Rightarrow \frac{{\rm{\pi }}}{2} - {\rm{cose}}{{\rm{c}}^{ - 1}}\left( {\sqrt 5 } \right) = {\rm{\theta }}\)

\(\therefore {\rm{cose}}{{\rm{c}}^{ - 1}}\left( {\sqrt 5 } \right) = \frac{{\rm{\pi }}}{2} - {\rm{\theta }}\)