Evaluate Cosec (cos-1 8/17)

Let \( cos^{-1}\dfrac{8}{17} = y\)

cos y = \(\dfrac{8}{17} \) where y ∈ [0, \(\dfrac{\pi}{2}\)]

Now, we have to find

\( Cosec\;( cos^{-1}\dfrac{8}{17} )= cosec\;y\)

We know that,

sin² θ + cos² θ = 1

sin² θ = \( \sqrt{ (1 – cos^2 θ)}\)

So,

sin y = \( \sqrt{ (1 – cos^2 y)}\)

= \( \sqrt{ (1 –\dfrac{8}{17}^2)}\)

= \(\sqrt{ \dfrac{225}{289}}\)

= \( \dfrac{15}{17}\)

Hence,

\( Cosec\; y = \dfrac{1}{sin\;y}\)

= \( \dfrac{1}{(\dfrac{15}{17})}=\)  \( \dfrac{17}{15}\)

Therefore,

\( Cosec\;( cos^{-1}\dfrac{8}{17} )= \dfrac{17}{15}\)