Trigonometric Expression Evaluation: cot(theta) = sqrt(7)

This problem requires evaluating a specific trigonometric expression using the given value of cotangent.

Expression Evaluation Setup

We are provided with the value of \(\cot\theta\) and asked to compute the value of a fraction involving \(\text{cosec}^2\theta\) and \(\sec^2\theta\).

• Given: \(\cot\theta = \sqrt{7}\)
• Target Expression: \(\frac{\text{cosec}^2\theta-\sec^2\theta}{\text{cosec}^2\theta+\sec^2\theta}\)

Essential Trigonometric Identities

The solution relies on these standard trigonometric relationships:

• \(\text{cosec}^2\theta = 1 + \cot^2\theta\)
• \(\sec^2\theta = 1 + \tan^2\theta\)
• \(\tan\theta = \frac{1}{\cot\theta}\)

Step-by-Step Solving Approach

We can solve this problem using two primary methods.

Direct Substitution Method

This approach involves calculating the individual components first.

1. Calculate \(\cot^2\theta\): Since \(\cot\theta = \sqrt{7}\), squaring both sides gives: \(\cot^2\theta = (\sqrt{7})^2 = 7\)
2. Calculate \(\text{cosec}^2\theta\): Using the identity \(\text{cosec}^2\theta = 1 + \cot^2\theta\): \(\text{cosec}^2\theta = 1 + 7 = 8\)
3. Calculate \(\tan\theta\): From \(\cot\theta = \sqrt{7}\), we find \(\tan\theta\): \(\tan\theta = \frac{1}{\cot\theta} = \frac{1}{\sqrt{7}}\)
4. Calculate \(\sec^2\theta\): Using the identity \(\sec^2\theta = 1 + \tan^2\theta\): \(\sec^2\theta = 1 + \left(\frac{1}{\sqrt{7}}\right)^2 = 1 + \frac{1}{7} = \frac{7}{7} + \frac{1}{7} = \frac{8}{7}\)
5. Substitute these values into the target expression: \(\frac{\text{cosec}^2\theta-\sec^2\theta}{\text{cosec}^2\theta+\sec^2\theta} = \frac{8 - \frac{8}{7}}{8 + \frac{8}{7}}\)
6. Simplify the numerator and denominator: Numerator: \(8 - \frac{8}{7} = \frac{56}{7} - \frac{8}{7} = \frac{48}{7}\) Denominator: \(8 + \frac{8}{7} = \frac{56}{7} + \frac{8}{7} = \frac{64}{7}\)
7. Compute the final value: \(\frac{48/7}{64/7} = \frac{48}{64}\)
8. Simplify the resulting fraction: Divide both numerator and denominator by their greatest common divisor, 16: \(\frac{48 \div 16}{64 \div 16} = \frac{3}{4}\)

Simplification using cot^2(theta) Method

This method simplifies the expression algebraically before substituting values.

1. Rewrite the expression. Divide the numerator and denominator by \(\sec^2\theta\): \(\frac{\frac{\text{cosec}^2\theta}{\sec^2\theta} - \frac{\sec^2\theta}{\sec^2\theta}}{\frac{\text{cosec}^2\theta}{\sec^2\theta} + \frac{\sec^2\theta}{\sec^2\theta}} = \frac{\frac{\text{cosec}^2\theta}{\sec^2\theta} - 1}{\frac{\text{cosec}^2\theta}{\sec^2\theta} + 1}\)
2. Recall the identity \(\frac{\text{cosec}^2\theta}{\sec^2\theta} = \frac{1/\sin^2\theta}{1/\cos^2\theta} = \frac{\cos^2\theta}{\sin^2\theta} = \cot^2\theta\). Substitute this into the expression: \(\frac{\cot^2\theta - 1}{\cot^2\theta + 1}\)
3. Substitute the value \(\cot^2\theta = 7\) (calculated in the previous method): \(\frac{7 - 1}{7 + 1} = \frac{6}{8}\)
4. Simplify the fraction: \(\frac{6}{8} = \frac{3}{4}\)

Final Calculated Value

Both methods confirm that the value of the given trigonometric expression \(\frac{\text{cosec}^2\theta-\sec^2\theta}{\text{cosec}^2\theta+\sec^2\theta}\) when \(\cot\theta = \sqrt{7}\) is \(\frac{3}{4}\).