Question

If Tan\(\theta\) + Cot\(\theta\) = 4, find Tan\(^2\)\(\theta\) + Cot\(^2\)\(\theta\).

Hint:

Recognize algebraic patterns in trigonometry. If you see expressions like \(x+1/x\) and are asked for \(x^2+1/x^2\), the squaring method is the fastest approach.

Answer 1

Correct Answer: 14

Step 1: Understanding the Question:
We are given the sum of Tan\(\theta\) and Cot\(\theta\) and are asked to find the sum of their squares. This is an algebraic manipulation problem.
Step 2: Key Formula or Approach:
We will use the algebraic identity: \( (a+b)^2 = a^2 + b^2 + 2ab \).
We also use the trigonometric identity: \( \text{Cot}\theta = \frac{1}{\text{Tan}\theta} \), which means \( \text{Tan}\theta \cdot \text{Cot}\theta = 1 \).
Step 3: Detailed Explanation:
Let the given equation be: \[ \text{Tan}\theta + \text{Cot}\theta = 4 \] Square both sides of the equation: \[ (\text{Tan}\theta + \text{Cot}\theta)^2 = 4^2 \] Expand the left side using the identity \( (a+b)^2 \): \[ \text{Tan}^2\theta + \text{Cot}^2\theta + 2(\text{Tan}\theta)(\text{Cot}\theta) = 16 \] We know that \( \text{Tan}\theta \cdot \text{Cot}\theta = 1 \). Substitute this value into the equation: \[ \text{Tan}^2\theta + \text{Cot}^2\theta + 2(1) = 16 \] \[ \text{Tan}^2\theta + \text{Cot}^2\theta + 2 = 16 \] Now, isolate the term we want to find by subtracting 2 from both sides: \[ \text{Tan}^2\theta + \text{Cot}^2\theta = 16 - 2 \] \[ \text{Tan}^2\theta + \text{Cot}^2\theta = 14 \] Step 4: Final Answer:
The value of Tan\(^2\)\(\theta\) + Cot\(^2\)\(\theta\) is 14.