Question

Prove that:tan²A cos² A=1-cos²A​

Answer 1

The proof involves using the Pythagorean trigonometric identity to express sin²A in terms of cos²A, and then using the definition of tangent to show that tan²A cos²A equals 1 - cos²A.

Step-by-step explanation:

To prove that tan²A cos²A equals 1 - cos²A, we need to manipulate the trigonometric identities. Let's start with the Pythagorean identity, which states:

sin²A + cos²A = 1

From this identity, we can express sin²A in terms of cos²A:

sin²A = 1 - cos²A

Now let's consider the definition of the tangent function in terms of sine and cosine:

tanA = sinA / cosA

If we square the tangent function, we get:

tan²A = sin²A / cos²A

Using the expression we found for sin²A, we substitute it into the equation for tan²A:

tan²A = (1 - cos²A) / cos²A

We then multiply both sides by cos²A to get:

tan²A cos²A = 1 - cos²A, which is what we were asked to prove.