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

3 min read Jun 16, 2024
(1+tan^2x)(1-sin^2x)=1

Proving the trigonometric identity: (1 + tan²x)(1 - sin²x) = 1

This article will provide a step-by-step proof of the trigonometric identity: (1 + tan²x)(1 - sin²x) = 1.

Understanding the Identities Involved

Before diving into the proof, let's recall some fundamental trigonometric identities:

  • Pythagorean Identity: cos²x + sin²x = 1
  • Tangent Identity: tan x = sin x / cos x

Proof:

  1. Start with the left-hand side of the equation: (1 + tan²x)(1 - sin²x)

  2. Substitute tan²x with sin²x/cos²x using the tangent identity: (1 + sin²x/cos²x)(1 - sin²x)

  3. Simplify by finding a common denominator for the first term: ((cos²x + sin²x)/cos²x)(1 - sin²x)

  4. Apply the Pythagorean Identity (cos²x + sin²x = 1) to simplify the first term: (1/cos²x)(1 - sin²x)

  5. Distribute the 1/cos²x: (1/cos²x) - (sin²x/cos²x)

  6. Recognize that sin²x/cos²x is equal to tan²x: (1/cos²x) - tan²x

  7. Use the Pythagorean Identity again (1 + tan²x = sec²x) to rewrite the first term: sec²x - tan²x

  8. Finally, apply the Pythagorean Identity (sec²x - tan²x = 1) to obtain the right-hand side of the equation: 1

Therefore, we have proven that (1 + tan²x)(1 - sin²x) = 1.

Conclusion

This proof demonstrates the interconnectedness of various trigonometric identities. By applying these identities in a systematic manner, we can simplify complex expressions and arrive at desired results. This particular identity is valuable in manipulating trigonometric equations and solving problems involving trigonometric functions.

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