Prove the following identity: (sqrt (1 - (cos^2 ) t ) * sqrt ((sec^2) t - 1)) / cos t = (tan^2) t

We have to prove: (sqrt (1 - (cos t)^2) * sqrt ((sec t
)^2) - 1)) / cos t = (tan t)^2

(sqrt (1 - (cos t)^2) * sqrt
((sec t )^2) - 1)) / cos t

=> (sqrt (sin t)^2 * sqrt
( 1/ (cos t)^2) - 1) / cos t

=> ((sin t) * sqrt ( 1/
(cos t)^2 - (cos t)^2/ (cos t)^2) / cos t

=> ((sin
t) * sqrt ( (1 - (cos t)^2) / (cos t)^2)/ cos t

=>
[(sin t * sqrt (1/ (cos t)^2)/ (cos t)] / cos t

=>
[(sin t * sqrt (( sin t)^2)/ (cos t)] / cos t

=>
[(sin t * sqrt (( sin t)^2)]/ (cos t)^2

=> [(sin t *
sin t]/ (cos t)^2

=> ( sin t )^2 / (cos t
)^2

=> (tan t)^2

We
obtain the right hand side by manipulating the terms of the left hand
side.

Therefore we prove that (sqrt (1 - (cos
t)^2) * sqrt ((sec t )^2) - 1)) / cos t = (tan
t)^2