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(1+tan^2x)cos^2x=1

2 min read Jun 16, 2024
(1+tan^2x)cos^2x=1

Proving the Trigonometric Identity: (1 + tan²x)cos²x = 1

This article will demonstrate the proof of the trigonometric identity: (1 + tan²x)cos²x = 1.

Understanding the Identity

This identity relates the tangent, cosine, and the constant value 1. It's essential to remember the following trigonometric definitions:

  • tan x = sin x / cos x
  • cos²x + sin²x = 1

Proof

  1. Start with the left-hand side of the equation: (1 + tan²x)cos²x

  2. Substitute tan²x with sin²x/cos²x: (1 + sin²x/cos²x)cos²x

  3. Combine the terms inside the parentheses: (cos²x + sin²x)/cos²x * cos²x

  4. Apply the Pythagorean Identity (cos²x + sin²x = 1): 1/cos²x * cos²x

  5. Simplify by canceling out cos²x: 1

  6. Therefore, the left-hand side equals the right-hand side: (1 + tan²x)cos²x = 1

Conclusion

We have successfully proven the trigonometric identity (1 + tan²x)cos²x = 1 by using the definitions of tangent and cosine, as well as the Pythagorean Identity. This identity is a useful tool in simplifying trigonometric expressions and solving trigonometric equations.

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