If x = m secA + n tanA and y = m tanA + n secA, then what is x 2 - y 2equal to ?
m 2 - n 2
Finding \(x^2 - y^2\) with Trigonometric Expressions
The problem asks us to find the value of \(x^2 - y^2\) given the expressions for \(x\) and \(y\) in terms of constants \(m\), \(n\), and trigonometric functions \(\sec A\) and \(\tan A\).
We are given:
- \(x = m \sec A + n \tan A\)
- \(y = m \tan A + n \sec A\)
Step 1: Calculate \(x^2\)
We need to square the expression for \(x\):
\(x^2 = (m \sec A + n \tan A)^2\)
Using the algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = m \sec A\) and \(b = n \tan A\):
\(x^2 = (m \sec A)^2 + 2(m \sec A)(n \tan A) + (n \tan A)^2\)
\(x^2 = m^2 \sec^2 A + 2mn \sec A \tan A + n^2 \tan^2 A\)
Step 2: Calculate \(y^2\)
Next, we square the expression for \(y\):
\(y^2 = (m \tan A + n \sec A)^2\)
Using the same algebraic identity \((a+b)^2 = a^2 + 2ab + b^2\), where \(a = m \tan A\) and \(b = n \sec A\):
\(y^2 = (m \tan A)^2 + 2(m \tan A)(n \sec A) + (n \sec A)^2\)
\(y^2 = m^2 \tan^2 A + 2mn \tan A \sec A + n^2 \sec^2 A\)
Step 3: Calculate \(x^2 - y^2\)
Now we subtract \(y^2\) from \(x^2\):
\(x^2 - y^2 = (m^2 \sec^2 A + 2mn \sec A \tan A + n^2 \tan^2 A) - (m^2 \tan^2 A + 2mn \tan A \sec A + n^2 \sec^2 A)\)
Distribute the negative sign:
\(x^2 - y^2 = m^2 \sec^2 A + 2mn \sec A \tan A + n^2 \tan^2 A - m^2 \tan^2 A - 2mn \tan A \sec A - n^2 \sec^2 A\)
Notice that the term \(+ 2mn \sec A \tan A\) and \(- 2mn \tan A \sec A\) cancel each other out because multiplication is commutative (\(\sec A \tan A = \tan A \sec A\)).
The expression simplifies to:
\(x^2 - y^2 = m^2 \sec^2 A + n^2 \tan^2 A - m^2 \tan^2 A - n^2 \sec^2 A\)
Step 4: Group Terms and Apply Trigonometric Identity
Group the terms involving \(m^2\) and \(n^2\):
\(x^2 - y^2 = (m^2 \sec^2 A - m^2 \tan^2 A) + (n^2 \tan^2 A - n^2 \sec^2 A)\)
Factor out \(m^2\) from the first group and \(n^2\) from the second group:
\(x^2 - y^2 = m^2 (\sec^2 A - \tan^2 A) - n^2 (\sec^2 A - \tan^2 A)\)
We used \((n^2 \tan^2 A - n^2 \sec^2 A) = - (n^2 \sec^2 A - n^2 \tan^2 A) = - n^2 (\sec^2 A - \tan^2 A)\).
Recall the fundamental trigonometric identity:
\(\sec^2 \theta - \tan^2 \theta = 1\)
Using this identity for \(\theta = A\):
\(\sec^2 A - \tan^2 A = 1\)
Substitute this into our expression for \(x^2 - y^2\):
\(x^2 - y^2 = m^2 (1) - n^2 (1)\)
\(x^2 - y^2 = m^2 - n^2\)
Thus, the value of \(x^2 - y^2\) is \(m^2 - n^2\).
Revision Table: Key Trigonometric Identities
| Identity | Description |
|---|---|
| \(\sin^2 \theta + \cos^2 \theta = 1\) | The Pythagorean identity relating sine and cosine. |
| \(\sec^2 \theta - \tan^2 \theta = 1\) | The Pythagorean identity relating secant and tangent. Used in this problem. |
| \(\csc^2 \theta - \cot^2 \theta = 1\) | The Pythagorean identity relating cosecant and cotangent. |
| \(\sec \theta = \frac{1}{\cos \theta}\) | Reciprocal identity for secant. |
| \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) | Ratio identity for tangent. |
Additional Information: Algebraic Manipulation in Trigonometry
Solving problems involving trigonometric expressions often requires a good understanding of both trigonometric identities and basic algebraic techniques. In this problem, we used the algebraic identity for squaring a binomial, \((a+b)^2\), and then factored terms to apply a trigonometric identity, \(\sec^2 A - \tan^2 A = 1\).
- Squaring Binomials: Remember \((a+b)^2 = a^2 + 2ab + b^2\) and \((a-b)^2 = a^2 - 2ab + b^2\).
- Difference of Squares: The form \(x^2 - y^2\) in the question hints that the difference of squares identity \(a^2 - b^2 = (a-b)(a+b)\) might be useful in some problems, although here direct expansion and subtraction was more effective due to the cancellation of terms.
- Factoring: Identifying common factors, like \(m^2\) and \((\sec^2 A - \tan^2 A)\) in our steps, is crucial for simplifying expressions and revealing identities.
- Using Identities: Always look for opportunities to substitute trigonometric identities to simplify expressions. The Pythagorean identities involving squares (\(\sin^2 + \cos^2\), \(\sec^2 - \tan^2\), \(\csc^2 - \cot^2\)) are particularly useful when dealing with squared trigonometric functions.
Mastering these techniques helps in solving a wide range of trigonometric problems efficiently.
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