$ p \sin^2 \alpha + q \cos^2 \alpha = m $ and $ p \cos^2 \beta + q \sin^2 \beta = n $
\(\frac{n-q}{p-n}\)
To find the value of \( \cot^2 \beta \), we begin with the given expressions:
- \( p \sin^2 \alpha + q \cos^2 \alpha = m \)
- \( p \cos^2 \beta + q \sin^2 \beta = n \)
We need to express \( \cot^2 \beta \) in terms of the given parameters.
The expression for \( \cot^2 \beta \) can be derived from the trigonometric identity:
- \( \cot \beta = \frac{\cos \beta}{\sin \beta} \)
Thus,
\(\cot^2 \beta = \frac{\cos^2 \beta}{\sin^2 \beta}\)
We substitute the second given equation into the expression for \( \cos^2 \beta \) and \( \sin^2 \beta \):
- From \( p \cos^2 \beta + q \sin^2 \beta = n \), we know:
- \( \cos^2 \beta = \frac{n-q \sin^2 \beta}{p} \)
Using the identity \( \cos^2 \beta + \sin^2 \beta = 1 \), we express \( \cos^2 \beta \) in another form:
\(\cos^2 \beta = 1 - \sin^2 \beta\)
Thus, substituting this into the expression for \( \cot^2 \beta \):
- \( \cot^2 \beta = \frac{1 - \sin^2 \beta}{\sin^2 \beta} \)
Let \( \sin^2 \beta = x \); therefore, \( \cot^2 \beta = \frac{1-x}{x} \).
From \( p \cos^2 \beta + q \sin^2 \beta = n \), replace \( \cos^2 \beta \):
- \( p(1-x) + qx = n \)
This simplifies to:
\(p - px + qx = n\)
Rearranging it:
\(x(q - p) = n - p\)
So:
\(x = \frac{n-p}{q-p}\)
Substitute \( x \) back into the expression for \( \cot^2 \beta \):
- \( \cot^2 \beta = \frac{1-\frac{n-p}{q-p}}{\frac{n-p}{q-p}} = \frac{q-p-(n-p)}{n-p} = \frac{q-n}{n-p} \)
Thus, the required value of \( \cot^2 \beta \) is:
\(\frac{n-q}{n-p}\)
Therefore, the correct answer is \(\frac{n-q}{n-p}\).
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 ?
Important Questions from Trigonometry
(secθ + tanθ)/(secθ - tanθ) is equal to:
If tan 45°, cot θ then the value of θ, in radians is
ABC is a triangle If sin (A+B)/2 = √3/2, then the value of sin C/2 is
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The value of sin θ and cos θ when the angle θ is 0 ≤ θ ≤ 90°, is ______
