Sint/1-cost=cosect+cott

Start on the left side. - cot (t) + sin (t) 1 - cos (t) - ⁢ cott + sint 1 - ⁢ cost Multiply sin (t) 1 - cos (t) sint 1 - ⁢ cost by 1 + cos (t) 1 + cos (t) 1 + cost 1 + cost. - cot (t) + sin (t) 1 - cos (t) 1 + cos (t) 1 + cos (t) - ⁢ cott + sint 1 - ⁢ cost ⁢ 1 + cost 1 + cost Combine. - cot (t) + sin (t) (1 + cos (t)) (1 - cos (t - cot (t) + sin (t) + sin (t) cos (t) (1 - cos (t)) (1 + cos (t)) - ⁢ cott + sint + sint ⁢ cost 1 - ⁢ cost ⁢ 1 + cost)) (1 + cos (t)) -

cot (t) + sin (t) + sin (t) cos (t) 1 - cos2 (t) - ⁢ cott + sint + sint ⁢ cost 1 - ⁢ cos2 t Apply pythagorean identity. - cot (t) + sin (t) + sin (t) cos (t) sin2 (t) - ⁢ cott + sint + sint ⁢ cost sin2 t Write cot (t) cott in sines and cosines using the quotient identity. - cos (t) sin (t) + sin (t) + sin (t) cos (t) sin2 (t) - ⁢ cost sint + sint + sint ⁢ cost sin2 t Simplify. 1 sin (t) 1 sint Rewrite 1 sin (t) 1 sint as csc (t) csct. csc (t) csct Because the two sides have been shown to be equivalent, the equation is an identity. - cot (t) + sin (t) 1 - cos (t) = csc (t) - ⁢ cott + sint 1 - ⁢ cost = csct is an identity