Determine whether the given identity is true or false If fal

Determine whether the given identity is true or false. If false find an appropriate equivalent expression. sec (t) + tan (t)/cot (t) + cos (t) = sec^2 (t)

Solution

( sec t + tan t ) / (cot t + cos t ) = sec^2 t

proving left hand side

sec t = 1/ cos t

tan t = sin t / cos t

cot t = cos t / sin t

( 1/ cos t + sin t / cos t ) / ( cos t / sin t + cos t )

( 1 + sin t ) / cos t / ( cos t + cos t sin t ) / sin t

( 1 + sin t ) / cos t / cos t ( 1 + sin t ) / sin t

sin t / cos^2 t

which is not equal to right side hence the identity is false

if right side is equal to sec t tan t then it would be true