$ \cosec \theta - \sin \theta = p^3 $ and $ \sec \theta - \cos \theta = q^3 $
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We are given two trigonometric equations:
- \(\cosec \theta - \sin \theta = p^3\)
- \(\sec \theta - \cos \theta = q^3\)
We need to find the value of the expression \(p^4q^2 + p^2q^4\).
Simplifying Trigonometric Equations
First, let's simplify the equation for \(p^3\):
\(p^3 = \frac{1}{\sin \theta} - \sin \theta = \frac{1 - \sin^2 \theta}{\sin \theta}\)
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we get \(1 - \sin^2 \theta = \cos^2 \theta\).
Thus, \(p^3 = \frac{\cos^2 \theta}{\sin \theta}\).
Next, simplify the equation for \(q^3\):
\(q^3 = \frac{1}{\cos \theta} - \cos \theta = \frac{1 - \cos^2 \theta}{\cos \theta}\)
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we get \(1 - \cos^2 \theta = \sin^2 \theta\).
Thus, \(q^3 = \frac{\sin^2 \theta}{\cos \theta}\).
Calculating Powers of p and q
We need the terms \(p^4q^2\) and \(p^2q^4\). Let's find the necessary powers:
\(p^4 = (p^3)^{4/3} = \left( \frac{\cos^2 \theta}{\sin \theta} \right)^{4/3} = \frac{\cos^{8/3} \theta}{\sin^{4/3} \theta}\)
\(q^2 = (q^3)^{2/3} = \left( \frac{\sin^2 \theta}{\cos \theta} \right)^{2/3} = \frac{\sin^{4/3} \theta}{\cos^{2/3} \theta}\)
\(p^2 = (p^3)^{2/3} = \left( \frac{\cos^2 \theta}{\sin \theta} \right)^{2/3} = \frac{\cos^{4/3} \theta}{\sin^{2/3} \theta}\)
\(q^4 = (q^3)^{4/3} = \left( \frac{\sin^2 \theta}{\cos \theta} \right)^{4/3} = \frac{\sin^{8/3} \theta}{\cos^{4/3} \theta}\)
Evaluating the Target Expression
Now, calculate the two product terms:
\(p^4q^2 = \left( \frac{\cos^{8/3} \theta}{\sin^{4/3} \theta} \right) \left( \frac{\sin^{4/3} \theta}{\cos^{2/3} \theta} \right) = \cos^{(8/3 - 2/3)} \theta = \cos^2 \theta\)
\(p^2q^4 = \left( \frac{\cos^{4/3} \theta}{\sin^{2/3} \theta} \right) \left( \frac{\sin^{8/3} \theta}{\cos^{4/3} \theta} \right) = \sin^{(8/3 - 2/3)} \theta = \sin^2 \theta\)
Summing these terms:
\(p^4q^2 + p^2q^4 = \cos^2 \theta + \sin^2 \theta\)
Using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\), the value is 1.
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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