Question

(tanTheta+secTheta-1)/(tanTheta-secTheta+1) = (1+sinTheta)/(cosTheta)

Prove that the two expressions are equal.

Answer 1

tanTheta = sinTheta/cosTheta
and secTheta=1/cosTheta

substitute them

Answer 2

done substituting?

Answer 3

This is an identity.

Right side = 0

Left side:

= secθ - sinθtanθ - cosθ

Change tanθ to sinθ/cosθ

secθ - sin^2θ/cosθ - cosθ

Multiply the cosθ by cosθ/cosθ and change the secθ to 1/cosθ

(1/cosθ) - (sin^2θ/cosθ) - (cos^2θ/cosθ)

You now have a common denominator:

[1 - sin^2θ - cos^2θ] / [cosθ]

Factor out a negative in the numerator

[1 - (sin^2θ + cos^2θ)] / [cosθ]

sin^2θ + cos^2θ = 1

[1 - (1)] / [cosθ]

0 / [cosθ]

0

So LS = RS, QED.