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

Simplify using the identities:
Let's substitute the first identity into the left side of the equation:
(sec²x)(cos²x  1) = tan²x

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

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

Combine terms:
 Multiply both sides by cos²x: cos²x  1 = sin²x
 Rearrange: cos²x + sin²x = 1

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.