Question

Prove that:

(a)

(1 + tan A)^2 + (1 – tan A)^2
-------------------------------
(1 + cot A)^2 + (1 - cot A)^2
= tan^2 A

How would you like to proceed with this mathematical proof?
a) True
b) False

Answer 1

The given equation can be proved to be true by expanding and simplifying the expressions in the numerator and denominator, and using trigonometric identities.

Step-by-step explanation:

To prove the given equation, let's start by expanding the expressions in the numerator and denominator:

(1 + tan A)^2 + (1 - tan A)^2 = (1 + tan^2 A + 2tan A) + (1 + tan^2 A - 2tan A)

(1 + cot A)^2 + (1 - cot A)^2 = (1 + cot^2 A + 2cot A) + (1 + cot^2 A - 2cot A)

Next, we can simplify both expressions:

(2 + 2tan^2 A) + (2 + 2cot^2 A) = 4 + 2tan^2 A + 2cot^2 A

Using the trigonometric identity tan^2 A = 1 - cot^2 A, we can further simplify the equation:

4 + 2(1 - cot^2 A) + 2cot^2 A = 4 + 2 - 2cot^2 A + 2cot^2 A = 6

Therefore, the equation simplifies to:

(1 + tan A)^2 + (1 - tan A)^2 / (1 + cot A)^2 + (1 - cot A)^2 = 6

Since tan^2 A is defined as (1 - cos^2 A) / (cos^2 A), we can rewrite the equation as:

6 = (1 - cos^2 A) / (cos^2 A) = tan^2 A