tan(x)[tan(−x)+cot(−x)]=−sec 
2
 (x) using odd/seven, reciprocal,agebra,pythagorean,qutioint


We are asked to prove the trigonometric identity:

\[
\text{tan}(x) \left( \text{tan}(-x) + \text{cot}(-x) \right) = - \sec^2(x)
\]

To do this, we will use the **odd/even identities**, **reciprocal identities**, **algebra**, **Pythagorean identities**, and **quotient identities**.

### Step 1: Use Odd/Even Identities
First, recall that:
- \(\tan(-x) = -\tan(x)\) (since tangent is an odd function),
- \(\cot(-x) = -\cot(x)\) (since cotangent is also an odd function).

Thus, the left-hand side becomes:
\[
\text{tan}(x) \left( \text{tan}(-x) + \text{cot}(-x) \right) = \text{tan}(x) \left( -\text{tan}(x) - \text{cot}(x) \right)
\]

### Step 2: Use Reciprocal Identities
We know that:
- \(\cot(x) = \frac{1}{\tan(x)}\) (reciprocal identity for cotangent).

Substitute this into the equation:
\[
\text{tan}(x) \left( -\text{tan}(x) - \frac{1}{\text{tan}(x)} \right)
\]

### Step 3: Simplify Using Algebra
Distribute \(\text{tan}(x)\) over the terms inside the parentheses:
\[
= \text{tan}(x) \cdot (-\text{tan}(x)) + \text{tan}(x) \cdot \left(- \frac{1}{\text{tan}(x)} \right)
\]
This simplifies to:
\[
-\text{tan}^2(x) - 1
\]

### Step 4: Use Pythagorean Identity
We use the Pythagorean identity for tangent and secant:
\[
\text{tan}^2(x) + 1 = \sec^2(x)
\]
Thus, \(-\text{tan}^2(x) - 1\) becomes:
\[
-\sec^2(x)
\]

### Final Result:
We have shown that:
\[
\text{tan}(x) \left( \text{tan}(-x) + \text{cot}(-x) \right) = -\sec^2(x)
\]
which completes the proof.

### Do you want more details or have any questions?
Here are five related questions to further explore:
1. How can we apply even/odd identities to other trigonometric functions like sine and cosine?
2. What is the geometric interpretation of \(\sec(x)\) and \(\tan(x)\)?
3. How does the Pythagorean identity help in simplifying other trigonometric expressions?
4. Can we prove the identity using a different approach, such as converting everything to sine and cosine?
5. How do reciprocal and quotient identities interrelate in more complex trigonometric proofs?

**Tip**: Memorize the odd/even and reciprocal identities for quick simplifications during trigonometric proofs.

tan(x)[tan(−x) + cot(−x)] = −sec^2(x) using odd/even identities, reciprocal identities, algebra, Pythagorean identities, quotient identities

Explore how to prove the trigonometric identity tan(x)[tan(−x) + cot(−x)] = −sec^2(x) using odd/even identities, reciprocal identities, algebra, and the Pythagorean identity. This step-by-step guide is perfect for students in Grades 10-12 who are studying advanced trigonometry.

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