Sin(t-2π)/sec((π/2)-t)

Question

Find:

\(\cfrac{sin(t - 2 \pi)}{sec(\frac{\pi}{2}-t)} \)

Answer 1

\(\cfrac{sin(t - 2 \pi)}{sec(\frac{\pi}{2}-t)} \) = \(\cfrac{sin(-(2\pi-t))}{cosect}\) (∵ sec (π/2 - θ) = cosec θ)

=\(\cfrac{-sin(2\pi-t)}{cosec\, t} \) (∵ sin (-θ) = -sinθ)

= \(\cfrac{-(-sin\,t)}{cosec\, t} \)  (∵ sin (2π - θ) = -sinθ)

= sin t / cosec t

= sin t sin t   \((∵\cfrac{1}{cosecθ}=sinθ)\)

= sin² t

Hence, \(\cfrac{sin(t-2π)}{sec(\frac{\pi}2-t)}=sin^2 t\)