How do you verify the identity 3sec^2thetatan^2theta+1=sec^6theta-tan^6theta?

1 Answer
Oct 3, 2016

See below

Explanation:

3sec^2thetatan^2theta+1=sec^6theta-tan^6theta

Right Side=sec^6theta-tan^6theta
=(sec^2theta)^3-(tan^2theta)^3->use difference of two cubes formula

=(sec^2theta-tan^2theta) (sec^4theta+sec^2thetatan^2theta+tan^4theta)

=1 *(sec^4theta+sec^2thetatan^2theta+tan^4theta)

=sec^4theta+sec^2thetatan^2theta+tan^4theta

=sec^2theta sec^2 theta+sec^2thetatan^2theta+tan^2theta tan^2 theta

=sec^2theta(tan^2theta+1) +sec^2thetatan^2theta+tan^2theta(sec^2theta-1)

=sec^2thetatan^2theta+sec^2theta+sec^2thetatan^2theta+sec^2thetatan^2theta-tan^2theta

=sec^2thetatan^2theta+sec^2thetatan^2theta+sec^2thetatan^2theta+sec^2theta-tan^2theta

=3sec^2thetatan^2theta +1

= Left Side