Question

\(\text{The value of    } cosec20°tan60°-sec20° \text{is }\)

• A. \(0\)
• B. \(1\)
• C. \(2\)
• D. \(4\)
• E. \(6\)

Answer 1

The Correct Option is D

Given:

• Expression to evaluate: \( \csc 20° \tan 60° - \sec 20° \)

Step 1: Rewrite trigonometric functions in terms of sine and cosine: \[ \csc 20° = \frac{1}{\sin 20°}, \quad \tan 60° = \frac{\sin 60°}{\cos 60°}, \quad \sec 20° = \frac{1}{\cos 20°} \]

Step 2: Substitute known values: \[ \sin 60° = \frac{\sqrt{3}}{2}, \quad \cos 60° = \frac{1}{2} \] \[ \tan 60° = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \]

Step 3: Rewrite the original expression: \[ \frac{1}{\sin 20°} \cdot \sqrt{3} - \frac{1}{\cos 20°} \] \[ = \frac{\sqrt{3}}{\sin 20°} - \frac{1}{\cos 20°} \]

Step 4: Combine the terms: \[ = \frac{\sqrt{3} \cos 20° - \sin 20°}{\sin 20° \cos 20°} \]

Step 5: Recognize the numerator as a trigonometric identity: \[ \sqrt{3} \cos 20° - \sin 20° = 2 \sin (60° - 20°) = 2 \sin 40° \]

Step 6: Simplify the denominator using double-angle identity: \[ \sin 20° \cos 20° = \frac{1}{2} \sin 40° \]

Step 7: Final simplification: \[ \frac{2 \sin 40°}{\frac{1}{2} \sin 40°} = 4 \]

The correct answer is (D) 4.

Answer 2

\(cosec20.tan60-sec20\)

\(=\dfrac{1}{sin20}×\dfrac{sin60}{cos60}-\dfrac{1}{cos20}\)

\(=\dfrac{sin60.cos20-cos60.sin20}{sin20.cos60.cos20}\)

\(=\dfrac{sin(60-20)}{\dfrac{1}{2}.sin20.cos20}\)

\(=\dfrac{2.sin40}{sin20.cos20}\)

\(=\dfrac{2sin2(20)}{sin20.cos20}\)

\(=\dfrac{4sin20.cos20}{sin20.cos20}\)

\(=4\)