91Ó°ÊÓ

Understanding trigonometric identities is essential for solving and simplifying trigonometric expressions. These identities are formulas that relate the angles of a triangle to its sides and other angles. Some common trigonometric identities include:

• Sine and cosine: \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\)
• Tangent and cotangent: \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\)
• Cosecant and secant: \(\text{csc} \theta = \frac{1}{\text{sin} \theta}, \text{sec} \theta = \frac{1}{\text{cos} \theta}\)

In the given exercise, we encounter \(\text{cot} x\), which is the cotangent function and is defined as \(\text{cot} x = \frac{1}{\text{tan} x}\) or \(\text{cot} x = \frac{\text{cos} x}{\text{sin} x}\). Recognizing and manipulating these identities can transform complex expressions into more manageable forms.

Quadratic equations are polynomial equations of the second degree, generally in the form \(ax^2 + bx + c = 0\). The method to solve them includes factoring, using the quadratic formula, or completing the square.

In our exercise, after substituting \(y = \text{cot}^2 x\), we get \(y^2 + 3y + 2\), a standard quadratic equation. To factorize this, find two numbers that multiply to the constant term (2) and add to the coefficient of \(y\) (3). These numbers are 1 and 2. Hence, \(y^2 + 3y + 2\) can be factored into \((y + 1)(y + 2)\). Recognizing this pattern is key to simplifying complex trigonometric expressions.

The substitution method simplifies the problem by replacing a variable or function with another symbol or variable. This technique is particularly useful in trigonometry and algebra, where direct manipulation of the original variables is difficult.

In our case, we substituted \(y = \text{cot}^2 x\) to transform the trigonometric expression \(\text{cot}^4 x + 3 \text{cot}^2 x + 2\) into the simpler quadratic form \((y^2 + 3y + 2)\). Factorizing \(y^2 + 3y + 2\) into \((y + 1)(y + 2)\) made it easier to handle.

After factoring, we substitute back \(\text{cot}^2 x\) for \(y\) to revert to the trigonometric terms. The final factored expression is \((\text{cot}^2 x + 1)(\text{cot}^2 x + 2)\). Verifying this step by expanding ensures that our substitution and factorization are correct. This method greatly simplifies solving complex equations.