How do you verify the identity sqrt((sinthetatantheta)/sectheta)=abs(sintheta)sinθtanθsecθ=|sinθ|?

2 Answers
Aug 21, 2016

Show that LHS = RHS for all thetaθ

Explanation:

LHS simplifies to sqrt ((sin(theta).sin(theta)/cos(theta))/(1/cos(theta))  sin(θ).sin(θ)cos(θ)1cos(θ)using the definitions of tan and sec.
This simplifies to sqrt(sin(theta)^2sin(θ)2
and hence to sin (theta)sin(θ), where sin(theta)sin(θ) ≥ 0
and RHS = sin(theta)sin(θ), where sin(theta)sin(θ) ≥ 0
Thus LHS = RHS for all values ot thetaθ.

Aug 21, 2016

Square LHS and simplify - as below.

Explanation:

LHS = sqrt((sin theta tan theta)/sec theta)LHS=sinθtanθsecθ

Consider the square of the LHS:

LHS^2 = (sin theta tan theta)/sec thetaLHS2=sinθtanθsecθ

= sin thetaxxsin theta/cos theta xx cos theta=sinθ×sinθcosθ×cosθ

= sin thetaxxsin theta/cancel cos theta xx cancel cos theta
(cos theta !=0 -> θ != pi/2 +npi for all n in ZZ)

= sin^2 theta

:. LHS^2 = sin^2 theta

LHS = +- sin theta = abs sin theta

LHS = RHS