Question

Verify. SHOW ALL WORK. You should have 4 steps. You will earn 1 point per step shown correctly.

cos x (tan x + cot x) = CSC x

Show Your Work"

A) Step 1: cos x * (sin x/cos x + cos x/sin x) = 1/sin x
B) Step 2: (sin x + cos^2 x) / sin x = 1/sin x
C) Step 3: (sin x + 1 - sin^2 x) / sin x = 1/sin x
D) Step 4: (1 + cos^2 x) / sin x = 1/sin x

Answer 1

The task was to verify the trigonometric identity cos x (tan x + cot x) = csc x. Steps 1 through 4 were provided, but step 4 is incorrect. The correct version of step 4 should conclude that (sin x + cos²x) / sin x simplifies to 1/sin x, not (1 + cos²x) / sin x.

Step-by-step explanation:

The problem requires verifying the identity cos x (tan x + cot x) = csc x. Below are the verification steps showing all required work:

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

   This step involves expressing tan x and cot x in terms of sine and cosine.

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

   Step two simplifies the expression by combining the terms over a common denominator.

3. (sin x + 1 - sin^2 x) / sin x = 1/sin x

   In this step, cos^2 x is replaced by (1 - sin^2 x) using the Pythagorean identity.

4. (1 + cos^2 x) / sin x = 1/sin x

   Step four is incorrect because the algebra is not properly applied. Instead, it should read 1/sin x after canceling sin x in the numerator and the denominator in the previous step.