Question

Prove : sin(x) / 1- sin(x) + sin(x) / 1+sin(x) = 2tan(x) / cos(x)) : true

Answer 1

To prove the identity, sin(x) / (1 - sin(x)) + sin(x) / (1 + sin(x)) = 2tan(x) / cos(x), we find a common denominator, use the Pythagorean identity to simplify, and express sin(x) in terms of tan(x) to establish the given relation.

Step-by-step explanation:

To prove the given identity sin(x) / (1 - sin(x)) + sin(x) / (1 + sin(x)) = 2tan(x) / cos(x), we can use trigonometric identities.

First, we need to find a common denominator for the left-hand side fractions:

• Simplify the left-hand side: sin(x) / (1 - sin(x)) + sin(x) / (1 + sin(x)) becomes ((sin(x) * (1 + sin(x))) + (sin(x) * (1 - sin(x)))) / ((1 - sin^2(x)))
• Apply the Pythagorean identity: 1 - sin^2(x) = cos^2(x), and cancel common terms.
• The expression simplifies to 2sin(x) / cos^2(x).

Now, express the numerator sin(x) in terms of tan(x) and divide by cos(x), we get:

• 2sin(x) / cos^2(x) can be written as 2(sin(x)/cos(x)) / cos(x)
• Which further simplifies to 2tan(x) / cos(x), proving the original identity.

Using trigonometric identities such as the Pythagorean identity simplifies the proof and allows the terms to cancel out as needed.