$ p \sin^2 \alpha + q \cos^2 \alpha = m $ and $ p \cos^2 \beta + q \sin^2 \beta = n $
\(\frac{m-q}{p-m}\)
In this problem, we are given the equation:
\(p \sin^2 \alpha + q \cos^2 \alpha = m\)
We need to find the value of \(\tan^2\alpha\) using the given condition.
Let's start by manipulating the given equation:
- Use the identity \(\sin^2 \alpha + \cos^2 \alpha = 1\).
- Express \(\sin^2 \alpha\) in terms of \(\cos^2 \alpha\):
\(\sin^2 \alpha = 1 - \cos^2 \alpha\). - Substitute \(\sin^2 \alpha\) in the given equation:
\(p(1 - \cos^2 \alpha) + q \cos^2 \alpha = m\) - Simplify it:
\(p - p \cos^2 \alpha + q \cos^2 \alpha = m\)
\(p + (q - p)\cos^2 \alpha = m\) - Isolating \(\cos^2 \alpha\), we get:
\((q - p) \cos^2 \alpha = m - p\)
\(\cos^2 \alpha = \frac{m - p}{q - p}\) - Using \(\tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha}\), we can express it as:
\(\tan^2 \alpha = \frac{1 - \cos^2 \alpha}{\cos^2 \alpha}\)
- Now substitute for \(\cos^2 \alpha\) in \(\tan^2 \alpha\):
\(\tan^2 \alpha = \frac{1 - \frac{m-p}{q-p}}{\frac{m-p}{q-p}}\) - Simplifying the expression:
\(\tan^2 \alpha = \frac{(q-p) - (m-p)}{m-p}\)
\(\tan^2 \alpha = \frac{q-m}{m-p}\)
Thus, the correct expression for \(\tan^2 \alpha\) is \(\frac{m-q}{p-m}\).
Similar Questions
If \(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} = \text{cosec}\theta\), then what is \(\sin\theta\) equal to?
- If \(2 \tan\theta = \sec^2\theta-2\), where \(0 < \theta < \pi /2\), then what is \(\cot\theta\) equal to?
- What is \(\frac{\tan^2 \alpha}{\tan^2 \beta}\) equal to?
- If \(\alpha\) and \(\beta\) are complementary angles, then which one of the following is correct?
- What is \(\sec\theta\) equal to?
- What is \(\tan\theta\) equal to?
- What is the relation between \(p\) and \(q\)?
- What is $\tan\theta + \cot\theta$ equal to?
- Consider the following statements :
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.
Which of the statements given above is/are correct? What is \((1 + \tan\alpha \tan\beta)^2 + (\tan\alpha - \tan\beta)^2\) equal to ?
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