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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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