The given trigonometric identity (cosθ − sinθ)/(cosθ + sinθ) = (1 − tanθ)/(1 + tanθ) has been proved.
In the question, we are asked to prove the trigonometric identity (cosθ − sinθ)/(cosθ + sinθ) = (1 − tanθ)/(1 + tanθ).
Going by the left-hand side of the equation, we get:
(cosθ − sinθ)/(cosθ + sinθ)
Dividing the numerator and the denominator by cosθ, we get:
{(cosθ − sinθ)/cosθ}/{(cosθ + sinθ)/cosθ}
= {cosθ/cosθ - sinθ/cosθ}/{cosθ/cosθ + sinθ/cosθ}
= (1 - sinθ/cosθ)(1 + sinθ/cosθ).
Using the identity, tanθ = sinθ/cosθ, we get:
(1 - tanθ)/(1 + tan θ)
= The right-hand side of the equation.
Hence the identity has been proved.
Thus, the given trigonometric identity (cosθ − sinθ)(cosθ + sinθ) = (1 − tanθ)(1 + tanθ) has been proved.
Learn more about trigonometric identities at
https://brainly.com/question/7331447
#SPJ9
The provided question is incomplete. The complete question is:
"Prove the trigonometric identity (cosθ − sinθ)/(cosθ + sinθ) = (1 − tanθ)/(1 + tanθ)."
