(1+tan^2x)(1-sin^2x)

Jasonbradley

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Jun 16, 2024

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Simplifying the Trigonometric Expression (1 + tan²x)(1 - sin²x)

This article will guide you through simplifying the trigonometric expression (1 + tan²x)(1 - sin²x). We will use fundamental trigonometric identities to achieve a simpler form.

Utilizing Trigonometric Identities

1. Recall the Pythagorean Identity:
   The fundamental trigonometric identity states: 1 + tan²x = sec²x.

2. Recognize the Identity for Cosine: Another crucial identity is 1 - sin²x = cos²x.

3. Substitute the Identities: Now, substitute these identities into our original expression: (1 + tan²x)(1 - sin²x) = (sec²x)(cos²x)

4. Simplify Using Reciprocal Identity: Since secx = 1/cosx, we can rewrite the expression as: (sec²x)(cos²x) = (1/cos²x)(cos²x)

5. Final Simplification: The cos²x terms cancel out, leaving us with: (1/cos²x)(cos²x) = 1

Conclusion

Therefore, the simplified form of the trigonometric expression (1 + tan²x)(1 - sin²x) is 1. This demonstrates the power of utilizing trigonometric identities to simplify complex expressions and reveal their fundamental relationships.