Question:
If \( sin~A=\frac{3}{5} \) and A lies in the second quadrant, then \( \frac{tan~A-sec~A}{cot~A+cosec~A}= \)
If \( sin~A=\frac{3}{5} \) and A lies in the second quadrant, then \( \frac{tan~A-sec~A}{cot~A+cosec~A}= \)
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Always double check ASTC signs before performing substitutions:
• Quadrant I: All positive.
• Quadrant II: Sine and Cosecant positive.
• Quadrant III: Tangent and Cotangent positive.
• Quadrant IV: Cosine and Secant positive.
• Quadrant I: All positive.
• Quadrant II: Sine and Cosecant positive.
• Quadrant III: Tangent and Cotangent positive.
• Quadrant IV: Cosine and Secant positive.
Updated On: Jun 8, 2026
- \( \frac{2}{5} \)
- \( \frac{11}{15} \)
- \( \frac{4}{3} \)
- \( \frac{3}{2} \)
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The Correct Option is D
Solution and Explanation
Concept:
In the second quadrant (\( 90^{\circ} < A < 180^{\circ} \)), the trigonometric functions sine and cosecant are positive, while cosine, secant, tangent, and cotangent are strictly negative.
Step 1: Determining individual trigonometric ratios.
Given \( sin~A = \frac{3}{5} \), the adjacent side length of the corresponding right-angled triangle is \( \sqrt{5^2 - 3^2} = 4 \). Accounting for quadrant signs: \[ cos~A = -\frac{4}{5}, \quad sec~A = -\frac{5}{4} \] \[ tan~A = -\frac{3}{4}, \quad cot~A = -\frac{4}{3}, \quad cosec~A = \frac{5}{3} \]
Step 2: Substituting values into the given expression.
Numerator: \[ tan~A - sec~A = \left(-\frac{3}{4}\right) - \left(-\frac{5}{4}\right) = -\frac{3}{4} + \frac{5}{4} = \frac{2}{4} = \frac{1}{2} \] Denominator: \[ cot~A + cosec~A = -\frac{4}{3} + \frac{5}{3} = \frac{1}{3} \]
Step 3: Evaluating the final fraction.
\[ \text{Value} = \frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{1}{2}}{\frac{1}{3}} = \frac{3}{2} \]
Step 1: Determining individual trigonometric ratios.
Given \( sin~A = \frac{3}{5} \), the adjacent side length of the corresponding right-angled triangle is \( \sqrt{5^2 - 3^2} = 4 \). Accounting for quadrant signs: \[ cos~A = -\frac{4}{5}, \quad sec~A = -\frac{5}{4} \] \[ tan~A = -\frac{3}{4}, \quad cot~A = -\frac{4}{3}, \quad cosec~A = \frac{5}{3} \]
Step 2: Substituting values into the given expression.
Numerator: \[ tan~A - sec~A = \left(-\frac{3}{4}\right) - \left(-\frac{5}{4}\right) = -\frac{3}{4} + \frac{5}{4} = \frac{2}{4} = \frac{1}{2} \] Denominator: \[ cot~A + cosec~A = -\frac{4}{3} + \frac{5}{3} = \frac{1}{3} \]
Step 3: Evaluating the final fraction.
\[ \text{Value} = \frac{\text{Numerator}}{\text{Denominator}} = \frac{\frac{1}{2}}{\frac{1}{3}} = \frac{3}{2} \]
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