For the following two (02) items: Let $p\sin^2 \alpha+q\cos^2 \alpha=m$, $q\sin^2 \beta+p\cos^2\beta=n$; $p \neq m, n$ and $q \neq m, n$.
Trigonometric Equations Analysis
We are given two trigonometric equations:
- \(p\sin^2 \alpha+q\cos^2 \alpha=m\)
- \(q\sin^2 \beta+p\cos^2\beta=n\)
We are also told that the angles \(\alpha\) and \(\beta\) are complementary. This means their sum is \(90^\circ\) (or \(\frac{\pi}{2}\) radians).
Mathematically, \(\alpha + \beta = 90^\circ\).
Complementary Angle Condition
From the complementary angle relationship, we know:
- \(\sin \beta = \sin(90^\circ - \alpha) = \cos \alpha\)
- \(\cos \beta = \cos(90^\circ - \alpha) = \sin \alpha\)
Squaring both sides of these identities, we get:
- \(\sin^2 \beta = \cos^2 \alpha\)
- \(\cos^2 \beta = \sin^2 \alpha\)
Substituting into the Second Equation
Now, let's substitute these results into the second given equation (\(n\)):
Original Equation 2: \(q\sin^2 \beta+p\cos^2\beta=n\)
Substitute \(\sin^2 \beta = \cos^2 \alpha\) and \(\cos^2 \beta = \sin^2 \alpha\):
\(q(\cos^2 \alpha) + p(\sin^2 \alpha) = n\)
Rearranging the terms to match the format of Equation 1:
\(p\sin^2 \alpha + q\cos^2 \alpha = n\)
Comparing Equations for m and n
Now we have two expressions:
- From Equation 1: \(p\sin^2 \alpha+q\cos^2 \alpha=m\)
- From Equation 2 (after substitution): \(p\sin^2 \alpha+q\cos^2 \alpha=n\)
Since the left-hand sides of both expressions are identical (\(p\sin^2 \alpha+q\cos^2 \alpha\)), their right-hand sides must also be equal.
Therefore, we can conclude that \(m = n\).
Deriving the Final Relation
If \(m = n\), we can rearrange this equation to find the correct relationship among the options.
Subtracting \(n\) from both sides gives:
\(m - n = n - n\)
\(m - n = 0\)
This result matches the fourth option provided.
Similar Questions
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1. In a triangle ABC, if \(\sin A + \sin B + \sin C = \frac{3\sqrt{3}}{2}\), then the triangle can be equilateral.
2. In a triangle ABC, if \(\cos A + \cos B + \cos C = \frac{3}{2}\), then the triangle can be equilateral.
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Which of the statements is/are correct?
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