If `theta` be acute and `tan theta + cot theta = 2`, then the value of `tan^5 theta+ cot^10 theta`is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem, we start with the given equation:

**Step 1:** Write down the given equation.
\[ \tan \theta + \cot \theta = 2 \]

**Hint:** Remember that \(\cot \theta\) is the reciprocal of \(\tan \theta\).

**Step 2:** Rewrite \(\cot \theta\) in terms of \(\tan \theta\).
\[ \tan \theta + \frac{1}{\tan \theta} = 2 \]

**Hint:** Use the property that \(\cot \theta = \frac{1}{\tan \theta}\).

**Step 3:** Let \(x = \tan \theta\). The equation becomes:
\[ x + \frac{1}{x} = 2 \]

**Hint:** This is a common algebraic manipulation; substituting variables can simplify the equation.

**Step 4:** Multiply both sides by \(x\) to eliminate the fraction:
\[ x^2 + 1 = 2x \]

**Hint:** Be careful with the multiplication; ensure all terms are accounted for.

**Step 5:** Rearrange the equation to form a standard quadratic equation:
\[ x^2 - 2x + 1 = 0 \]

**Hint:** This can be factored or solved using the quadratic formula.

**Step 6:** Factor the quadratic:
\[ (x - 1)^2 = 0 \]

**Hint:** Recognize that this indicates a double root.

**Step 7:** Solve for \(x\):
\[ x - 1 = 0 \implies x = 1 \]

**Hint:** Since \(x = \tan \theta\), we find that \(\tan \theta = 1\).

**Step 8:** Find \(\cot \theta\):
\[ \cot \theta = \frac{1}{\tan \theta} = 1 \]

**Hint:** This confirms that both \(\tan \theta\) and \(\cot \theta\) are equal.

**Step 9:** Now, we need to find \(\tan^5 \theta + \cot^{10} \theta\):
\[ \tan^5 \theta + \cot^{10} \theta = 1^5 + 1^{10} \]

**Hint:** Substitute the values of \(\tan \theta\) and \(\cot \theta\) into the expression.

**Step 10:** Calculate the final result:
\[ 1^5 + 1^{10} = 1 + 1 = 2 \]

**Final Answer:**
\[ \tan^5 \theta + \cot^{10} \theta = 2 \]

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