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(2+tan^2x)/sec^2x-1

2 min read Jun 16, 2024
(2+tan^2x)/sec^2x-1

Simplifying the Expression (2 + tan²x) / (sec²x - 1)

This article will guide you through the process of simplifying the trigonometric expression (2 + tan²x) / (sec²x - 1). We'll utilize key trigonometric identities to arrive at a more concise and manageable form.

Understanding the Identities

Before we begin simplifying, let's recall some fundamental trigonometric identities:

  • Pythagorean Identity: 1 + tan²x = sec²x
  • Reciprocal Identity: secx = 1/cosx

Simplifying the Expression

  1. Simplify the denominator: Using the Pythagorean identity, we can replace sec²x - 1 with tan²x:

    (2 + tan²x) / (sec²x - 1) = (2 + tan²x) / tan²x

  2. Separate the terms: Divide each term in the numerator by the denominator:

    (2 + tan²x) / tan²x = 2/tan²x + tan²x/tan²x

  3. Simplify using reciprocal identities: Remember that tanx = sinx/cosx and cotx = cosx/sinx. Therefore, 1/tan²x = cot²x:

    2/tan²x + tan²x/tan²x = 2cot²x + 1

Therefore, the simplified form of the expression (2 + tan²x) / (sec²x - 1) is 2cot²x + 1.

Conclusion

By applying fundamental trigonometric identities, we have successfully simplified the complex expression. This demonstrates the power of these identities in manipulating trigonometric expressions to obtain simpler and more usable forms.

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