Final answer:
The trigonometric identity is proven by manipulating the terms separately with trigonometric identities and the Pythagorean Identity, ending with both terms being equivalent to sinθ + cosθ.
Step-by-step explanation:
To prove the trigonometric identity (cosθ / (1 - tanθ)) + (sinθ / (1 - cotθ)) = sinθ + cosθ, let's manipulate each term separately and use trigonometric identities.
For the first term, cosθ / (1 - tanθ), we know that tanθ = sinθ / cosθ. Therefore, 1 - tanθ becomes (cosθ - sinθ) / cosθ. So, we have:
cosθ / (cosθ - sinθ) = cosθ / (cosθ - sinθ) × (cosθ + sinθ) / (cosθ + sinθ)
Expanding the denominator we get cos2θ - sin2θ, which is a difference of squares and equals to 1 via the Pythagorean Identity (cos2θ + sin2θ = 1). Thus, we are left with cosθ + sinθ in the numerator.
Similarly, for the second term sinθ / (1 - cotθ), using cotθ = cosθ / sinθ. So, 1 - cotθ turns into (sinθ - cosθ) / sinθ. Following similar steps as above, by multiplying the numerator and denominator with (sinθ + cosθ), and using the Pythagorean Identity, we get sinθ + cosθ.
Combining these results:
(cosθ + sinθ) + (cosθ + sinθ) = sinθ + cosθ
Therefore, the identity is proven.