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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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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

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