sec6(x) sec(x) tan(x) - sec4(x) sec(x) tan(x)
= sec7(x) tan(x) - sec5(x) tan(x) Factor out sec5(x) tan(x) .
= sec5(x) tan(x) * ( sec2(x) - 1 ) Rewrite secant as 1/cosine
= sec5(x) tan(x) * ( 1/cos2(x) - 1 ) Get a common denominator and combine the fractions.
= sec5(x) tan(x) * ( 1/cos2(x) - cos2(x)/cos2(x) )
= sec5(x) tan(x) * ( [ 1 - cos2(x) ] /cos2(x) ) Rewite 1 - cos2(x) as sin2(x)
= sec5(x) tan(x) * ( sin2(x) /cos2(x) )
= sec5(x) tan(x) * tan2(x)
= sec5(x) tan3(x)