Question

verify identity ((1+cos3t)/sin3t)) + (sin3t/1+cos3t) = 2csc3t

Answer 1

\[\frac{ 1+\cos3t }{ \sin3t } + \frac{ \sin3t }{ 1+\cos3t } = 2 \csc3t\]

Answer 2

one way to do it is to combine the two fractions in the LHS.
\[\large \frac{1+\cos 3t}{\sin 3t}+\frac{\sin 3t}{1+\cos 3t}=\frac{(1+\cos 3t)^2 + (\sin 3t)^2}{\sin 3t (1+\cos 3t)}\]

Answer 3

a second way is to multiply the 2nd fraction before combining them
\[\large \frac{1+\cos 3t}{\sin 3t}+\frac{\sin 3t(1-\cos 3t)}{(1+\cos 3t)(1-\cos 3t)}\]