proving identities

Question

tan α - cosec α secan α (1-2 cos^2 α) = cotan α

Answer 1

tan α - cosec α secan α (1-2 cos^2 α)

=sinα/cosα-(1/sinα)(1/cosα)(1-cos²α-cos²α)

=sinα/cosα-(1/sinα)(1/cosα)(sin²α-cos²α)

=sin²α/sinαcosα-(sin²α-cos²α)/sinαcosα

=(sin²α-sin²α+cos²α)/sinαcosα

=cos²α/sinαcosα=cosαα/sinα=cotα

Answer 2

1-2 cos^2 α = 1 - 2 (1-sin^2 α) = -(1-2 sin^2 a)

tan α - cosec α secan α (1-2 cos^2 α) = (sin α/cos α) - (1 / sin α) * (1 /cos α) * -(1-2 sin^2 α)

= (sin α/cos α) + (1-2 sin^2 α)* (1 / sin α) * (1 /cos α)

Taking lowest common multiple of the two expressions (sin a * cos α) :

= (sin^2 α + 1 - 2 sin^2 α )/(sin α * cos α )

= (1 - sin^2 α )/(sin α * cos α )

= (cos ^2 α)/(sin α * cos α )

= (cos α)/(sin α)

=cotan α