Question

(csc^(2)x-1)/(1+sinx) simplif

Answer 1

[AI-generated answer — Bot GPT 3.5]

Let's simplify the given expression step by step:

(csc^(2)x - 1) / (1 + sinx)

First, we know that csc(x) = 1/sinx, so csc^(2)x = (1/sinx)^(2) = 1/sin^(2)x.

Now, we can replace csc^(2)x in the expression:

(1/sin^(2)x - 1) / (1 + sinx)

Next, we need to find a common denominator for the fractions in the numerator:

(1 - sin^(2)x) / sin^(2)x * (1 + sinx)

Now, we can simplify the numerator by using the trigonometric identity sin^(2)x + cos^(2)x = 1:

(cos^(2)x) / sin^(2)x * (1 + sinx)

Now, we can simplify further by using the identity cosx = √(1 - sin^(2)x):

(1 - sin^(2)x) / sin^(2)x * (1 + sinx)

(cos^(2)x) / sin^(2)x * (1 + sinx)

= cosx / sinx * (1 + sinx)

= cosx + cosx sinx / sinx

= cosx + cosx

= 2cosx

Therefore, the simplified expression is 2cosx.