Final answer:
To verify the identity 1 - sin x/cos x = cos x/(1 + sin x), we manipulated both sides to achieve a common denominator, utilized the Pythagorean identity, and showed that both sides of the equation are equal, thereby verifying the trigonometric identity.
Step-by-step explanation:
To verify the identity 1 - sin x/cos x = cos x/(1 + sin x), we start by getting a common denominator for the left side of the equation:
1 - sin x/cos x = (cos x - sin x)/cos x
Next, we transform the right side by multiplying the numerator and the denominator by (1 - sin x) to get a single fraction:
cos x/(1 + sin x) * (1 - sin x)/(1 - sin x) = (cos x - sin x cos x)/(1 - sin² x)
Using the Pythagorean identity sin² x + cos² x = 1, we can express (1 - sin² x) as cos² x:
(cos x - sin x cos x)/cos² x = (cos x - sin x)/cos x
The right side now matches the left side, thus verifying the trigonometric identity.