Simplify cosx/secx + sinxcosx/cotx

Answer 1

To simplify the expression, let's convert all the trigonometric functions into sine and cosine functions.

Recall the following trigonometric identities:
1. sec(x) = 1/cos(x)
2. cot(x) = 1/tan(x) = cos(x)/sin(x)

Using these identities, we can rewrite the expression as follows:
cos(x)/sec(x) + sin(x)cos(x)/cot(x)

= cos(x)/(1/cos(x)) + sin(x)cos(x)/(cos(x)/sin(x))

= cos(x) * cos(x) + sin(x)cos(x) * (sin(x)/cos(x))

= cos^2(x) + sin(x)cos(x) * sin(x)/cos(x)

= cos^2(x) + sin^2(x)

Finally, using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can simplify the expression to:

cos^2(x) + sin^2(x) = 1

Therefore, the simplified form of cos(x)/sec(x) + sin(x)cos(x)/cot(x) is 1.