Finding the Value of tan A from tan A + cot A = 2

The problem provides the equation \(\tan \text{A} + \cot \text{A} = 2\), with the condition that A is an acute angle (\(0 < \text{A} < 90^\circ\)). We need to find the value of \(\tan \text{A}\) first.

We know that \(\cot \text{A}\) is the reciprocal of \(\tan \text{A}\). So, we can write the equation in terms of \(\tan \text{A}\):

\(\tan \text{A} + \frac{1}{\tan \text{A}} = 2\)

Let's assume \(x = \tan \text{A}\). The equation becomes:

\(x + \frac{1}{x} = 2\)

To solve for \(x\), multiply the entire equation by \(x\) (since \(0 < \text{A} < 90^\circ\), \(\tan \text{A}\) is positive, so \(x \neq 0\)):

\(x \cdot x + x \cdot \frac{1}{x} = 2 \cdot x\)

\(x^2 + 1 = 2x\)

Rearrange the terms to form a quadratic equation:

\(x^2 - 2x + 1 = 0\)

This is a perfect square trinomial. It can be factored as:

\((x - 1)^2 = 0\)

Taking the square root of both sides:

\(x - 1 = 0\)

Solving for \(x\):

\(x = 1\)

Since we set \(x = \tan \text{A}\), this means:

\(\tan \text{A} = 1\)

Given the condition \(0 < \text{A} < 90^\circ\), the angle A for which \(\tan \text{A} = 1\) is \(45^\circ\). This confirms our value of \(\tan \text{A}\) is correct for the given range.

Evaluating the Series: \(\tan^2 \text{A} + \tan^3 \text{A} + \tan^4 \text{A} + \dots + \tan^n \text{A}\)

Now that we know \(\tan \text{A} = 1\), we can substitute this value into the given series:

The series is \(\tan^2 \text{A} + \tan^3 \text{A} + \tan^4 \text{A} + \dots + \tan^n \text{A}\).

Substitute \(\tan \text{A} = 1\) into each term:

• \(\tan^2 \text{A} = (1)^2 = 1\)
• \(\tan^3 \text{A} = (1)^3 = 1\)
• \(\tan^4 \text{A} = (1)^4 = 1\)
• ...
• \(\tan^n \text{A} = (1)^n = 1\)

So, the series becomes a sum of ones:

\(1 + 1 + 1 + \dots + 1\)

Calculating the Sum of the Series

We need to find the number of terms in the series \(1 + 1 + 1 + \dots + 1\). The original series was \(\tan^2 \text{A} + \tan^3 \text{A} + \tan^4 \text{A} + \dots + \tan^n \text{A}\). The powers of \(\tan \text{A}\) range from 2 to \(n\).

To count the number of terms, we can subtract the starting power from the ending power and add 1:

Number of terms = (Ending power) - (Starting power) + 1

Number of terms = \(n - 2 + 1\)

Number of terms = \(n - 1\)

Since each term in the series is equal to 1, the sum of the series is the number of terms multiplied by 1.

Sum = (Number of terms) \(\times 1\)

Sum = \((n - 1) \times 1\)

Sum = \(n - 1\)

Therefore, the value of \(\tan^2 \text{A} + \tan^3 \text{A} + \tan^4 \text{A} + \dots + \tan^n \text{A}\) is \(n - 1\).

Revision Table: Key Concepts

Additional Information: Related Trigonometry and Series Ideas

This problem combines concepts from trigonometry and series. Understanding the relationship between trigonometric functions and how to sum a series of terms are crucial.

• Trigonometric Reciprocal Identities: \(\cot \text{A} = \frac{1}{\tan \text{A}}\), \(\sec \text{A} = \frac{1}{\cos \text{A}}\), \(\csc \text{A} = \frac{1}{\sin \text{A}}\). These are fundamental for simplifying expressions and equations.
• Special Angle Values: Knowing the values of trigonometric functions for angles like \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) is very helpful. In this case, \(\tan 45^\circ = 1\) was key.
• Solving Trigonometric Equations: Equations like \(\tan \text{A} + \cot \text{A} = 2\) can often be transformed into algebraic equations (like the quadratic equation we solved) using identities. Always check the domain of the angle (like \(0 < \text{A} < 90^\circ\)) to determine the valid solutions for A.
• Finite Series: A finite series is the sum of a finite number of terms, like \(a_1 + a_2 + \dots + a_n\). If all terms are the same constant, the sum is simply the number of terms times the constant value. Counting the number of terms correctly is essential.