How do you verify the identify sinθtanθ+cosθ=secθ?

3 Answers
Mar 9, 2018

See the proof below

Explanation:

We need

tanθ=sinθcosθ

sin2θ+cos2θ=1

secθ=1cosθ

Therefore,

LHS=sinθtanθ+cosθ

=sinθsinθcosθ+cosθ

=sin2θ+cos2θcosθ

=1cosθ

=secθ

=RHS

QED

Mar 9, 2018

Apply the identities tan(θ)=sinθcosθ and secθ=1cosθ along with the Pythagorean theorem.

Explanation:

Apply the identity tan(θ)=sinθcosθ:
L.H.S.=sinθsinθcosθ+cosθ
=sin2θcosθ+cosθ
=sin2θ+cos2θcosθ

Apply the Pythagorean theorem sin2θ+cos2θ=1
=sin2θ+cos2θcosθ
=1cosθ

By the definition of secants 1cosθ=secθ:
=secθ
=R.H.S

Mar 9, 2018

see explanation

Explanation:

using the trigonometric identities

xtanθ=sinθcosθ and secθ=1cosθ

xsin2θ+cos2θ=1

Consider the left side

sinθtanθ+cosθ

=sinθ×sinθcosθ+cosθ

=sin2θcosθ+cos2θcosθcommon denominator cosθ

=sin2θ+cos2θcosθ

=1cosθ=secθ= right side verified