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

Jasonbradley

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Jun 16, 2024

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Solving the Trigonometric Equation: (tan²x + 1)(cos²x - 1) = -tan²x

This article will guide you through the process of solving the trigonometric equation (tan²x + 1)(cos²x - 1) = -tan²x. We'll use trigonometric identities and algebraic manipulation to simplify the equation and find the solutions.

Understanding the Equation

The given equation involves the trigonometric functions tangent (tan) and cosine (cos). To solve it, we need to utilize the following fundamental trigonometric identities:

• tan²x + 1 = sec²x
• cos²x + sin²x = 1
• secx = 1/cosx

Solving the Equation

1. Simplify using the identities:

   Let's substitute the first identity into the left side of the equation:

   (sec²x)(cos²x - 1) = -tan²x

2. Simplify further:

   • Expand the left side: sec²x * cos²x - sec²x = -tan²x
   • Substitute secx = 1/cosx: (1/cos²x) * cos²x - (1/cos²x) = -tan²x
   • Simplify: 1 - (1/cos²x) = -tan²x

3. Substitute for tan²x:

   • Recall that tan²x = sin²x / cos²x
   • Substitute this into the equation: 1 - (1/cos²x) = -sin²x/cos²x

4. Combine terms:

   • Multiply both sides by cos²x: cos²x - 1 = -sin²x
   • Rearrange: cos²x + sin²x = 1

5. The Solution:

   The equation now becomes the fundamental trigonometric identity cos²x + sin²x = 1. This equation is true for all values of x.

Conclusion

Therefore, the equation (tan²x + 1)(cos²x - 1) = -tan²x is an identity. It holds true for all values of x. There are no specific solutions to find as the equation is always true.