Evaluate : cosec ⁻¹ (- 2 3 3 )+ ⁻¹ (- 3 3 )+ ⁻¹ 2+ ⁻¹ (- 1 2 )- ⁻¹ ( 2 2 )

Step-by-step solution

The expression is E = cosec ⁻¹ (- 2 3 3 ) + ⁻¹ (- 3 3 ) + ⁻¹(2) + ⁻¹ (- 1 2 ) - ⁻¹ ( 2 2 ) . Simplify each term: 1. cosec ⁻¹ (- 2 3 ) = - cosec ⁻¹ ( 2 3 ) = - 3 . 2. ⁻¹ (- 1 3 ) = - ⁻¹ ( 1 3 ) = - 6 . 3. ⁻¹(2) = 3 . 4. ⁻¹ (- 1 2 ) = - ⁻¹ ( 1 2 ) = - 3 = 2 3 . 5. ⁻¹ ( 1 2 ) = 4 . Summing these values: E = - 3 - 6 + 3 + 2 3 - 4 E = - 6 + 2 3 - 4 Finding a common denominator of 12: E = -2 + 8 - 3 12 = 3 12 = 4 . Answer: 4