Use fundamental identities to write tan t sec2t in terms of sint only. Solution tan^2 t sec^2 t tan t = sin t / cos t sec t = 1/ cos t so , we can write tan^2 t sec^2 t = sin^2 t / cos^2 t * 1/ cos^2 t = sin^2 t / cos^4 t cos^4 t = cos^2 t * cos^2 t cos^2 t = ( 1- sin^2 t ) cos^4 t = ( 1 - sin^2 t )^2 = 1 - 2sin^2 t + sin^4 t hence, we can write tan^2 t sec^2 t = (sin^2 t )/ ( sin^4 t - 2sin^2 t + 1 ).

1. Use fundamental identities to write tan t sec2t in terms of sint only.
Solution
tan^2 t sec^2 t
tan t = sin t / cos t
sec t = 1/ cos t
so , we can write
tan^2 t sec^2 t = sin^2 t / cos^2 t * 1/ cos^2 t
= sin^2 t / cos^4 t
cos^4 t = cos^2 t * cos^2 t
cos^2 t = ( 1- sin^2 t )
cos^4 t = ( 1 - sin^2 t )^2
= 1 - 2sin^2 t + sin^4 t
hence, we can write
tan^2 t sec^2 t = (sin^2 t )/ ( sin^4 t - 2sin^2 t + 1 )