Question

If \(\tan A+\cot A=3\) then \(\tan ^{3} A+\cot ^{3} A=\) (a) 27 (b) 24 (c) 9 (d) 18

Short answer

Answer: None of the given options (a), (b), (c), (d) are correct.

Step-by-step solution

$$\tan A + \frac{1}{\tan A} = 3$$ #Step 2: Find a common denominator# To combine the terms on the left side, find a common denominator. In this case, the common denominator is \(\tan A\).

$$\frac{\tan^2 A + 1}{\tan A} = 3$$ #Step 3: Eliminate the fraction# To eliminate the fraction, multiply both sides of the equation by \(\tan A\).

$$\tan^2 A + 1 = 3 \tan A$$ #Step 4: Re-arrange to a quadratic form# Re-arrange the equation to form a quadratic equation in terms of \(\tan A\).

$$\tan^2 A - 3\tan A + 1 = 0$$ #Step 5: Solve the quadratic equation# Solve the quadratic equation for \(\tan A\). Here, notice that this quadratic doesn't have a simple solution (one that is easy to factorize). Nonetheless, we can substitute given expression to simplify it. #Step 6: Express the desired expression in terms of given expression# We need to find the expression \(\tan^3 A + \cot^3 A\). Let's substitute the value we found in step 1 for the cotangent.

$$\tan^3 A + \left(\frac{1}{\tan A}\right)^3 = \tan^3 A + \frac{1}{\tan^3 A}$$ #Step 7: Find a common denominator for the expression we need to find# To combine the terms, find a common denominator. In this case, the common denominator is \(\tan^3 A\).

$$\frac{\tan^6 A + 1}{\tan^3 A}$$ #Step 8: Use the given expression to simplify the desired expression# We are given that \(\tan A + \cot A = 3\). We will use this expression to simplify the desired expression in step 7. We know that, \((\tan A + \cot A)^3 = 3^3 = 27\) Now, let's expand \((\tan A + \cot A)^3\) and use the substitution from step 1.

$$(\tan A + \frac{1}{\tan A})^3 = 27$$

$$\frac{(\tan^3 A + 3\tan A)(\tan^3 A + 1)}{\tan^3 A} = 27$$

$$\frac{\tan^6 A + 3\tan^3 A + 2\tan^3 A + \cancel{3\tan A}}{\cancel{\tan^3 A}} = 27$$ #Step 9: Simplify and find the desired expression# Now we can simplify the expression and compare with our desired expression from step 7.

$$\frac{\tan^6 A + 5\tan^3 A + \cancel{3\tan A}}{\cancel{\tan^3 A}} = 27$$ Notice that \(\tan^6 A + 5\tan^3 A = \tan^6 A + 1 + (\text{some extra terms})\). Our desired expression is \(\frac{\tan^6 A + 1}{\tan^3 A}\). Since the extra terms are positive, they will make the overall fraction larger than our desired expression. Therefore, \(\tan^3 A + \cot^3 A\) is not equal to any of the given options. So, none of the given options (a), (b), (c), (d) are correct.

Key concepts

Tangent and Cotangent Relationship

In trigonometry, the tangent and cotangent functions are inverses, which creates an interesting relationship. The tangent of an angle, \( \tan A \), is defined as the ratio of the side opposite the angle to the adjacent side in a right triangle. On the other hand, the cotangent, \( \cot A \), is the reciprocal of the tangent, given by \( \cot A = \frac{1}{\tan A} \).

• If the sum of the tangent and cotangent is given, it provides valuable insights into the angle's properties.
• This relationship can be used to derive certain expressions, such as \( \tan^3 A + \cot^3 A \).

An interesting fact about these functions is that if you know one, you can quickly find the other. This reciprocal nature simplifies many problems in trigonometry.In our initial problem, the equation \( \tan A + \cot A = 3 \) lays the foundation for the mathematical exploration that follows.

Quadratic Equations in Trigonometry

Quadratic equations are common in trigonometric problems, often involving expressions like \( \tan^2 A \) and \( \tan A \). They typically take the form \( ax^2 + bx + c = 0 \) where a, b, and c are constants. Once we rearrange the expression \( \tan^2 A - 3\tan A + 1 = 0 \), we recognize this as a quadratic equation in terms of \( \tan A \). Solving such equations usually involves factoring, using the quadratic formula, or completing the square, all methods designed to find the value of \( \tan A \).

• Understanding how to manipulate quadratic expressions is essential for tackling higher-level trigonometric problems.
• In trigonometry, these solutions are often substituted back into expressions to find the values of related trigonometric functions.

Quadratic solutions give us potential angles or trigonometric values that satisfy the initial conditions or relationships, such as the given sum of \( \tan A + \cot A \). This, in turn, behaves as a stepping stone to solving complex trigonometric expressions.

Simplifying Trigonometric Expressions

Simplifying expressions in trigonometry involves using identities and algebraic manipulation to derive simpler or more insightful forms of equations. In our example, simplifying involves transforming \( \tan^3 A + \cot^3 A \) using the identity \( (\tan A + \cot A)^3 = \tan^3 A + 3\tan A\cot A + 1 \). Here, recognizing identities and simplifying the expression helps clarify the relationship between these trigonometric terms under the given condition \( \tan A + \cot A = 3 \).

• This enables us to compare expressions with known values or ranges.
• Helps identify erroneous options quickly when verifying multiple-choice answers.

While simplifying is often straightforward, it requires a good grasp of fundamental trigonometric identities.In this problem, leveraging simplicity allows determining if \( \tan^3 A + \cot^3 A \) matches any of the given options, involving expanding the cube and canceling terms effectively for clarity.