Prove the following identity: (tan^4) t + (tan^2) t + 1 = (1 - (sin^2) t * (cos^2) t) / (cos^4) t
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We have to prove that (tan t)^4 + (tan t)^2 + 1 = (1 - (sin t)^2* (cos t)^2) / (cos t)^4
The left hand side:
(tan t)^4 + (tan t)^2 + 1
=> (sin t)^4 / (cos t)^4 + (sin t)^2 / (cos t)^2 + 1
=> (sin t)^4 / (cos t)^4 + (cos t)^2*(sin t)^2 / (cos t)^4 + (cos t^4) / (cos t)^4
=> [(sin t)^4 + (cos t)^2*(sin t)^2 + (cos t^4)] / (cos t)^4
=> [ (sin t)^4 + (cos t )^4 + (cos t)^2*(sin t)^2]/(cos t)^4
=> [ ((sin t)^2 + (cos t)^2)^2 - 2*(cos t)^2*(sin t)^2 +(cos t)^2*(sin t)^2]/(cos t)^4
=> [ 1 - (cos t)^2*(sin t)^2]/(cos t)^4
which is the right hand side.
Therefore we have proved that (tan t)^4 + (tan t)^2 + 1 = (1 - (sin t)^2* (cos t)^2) / (cos t)^4
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