The value of 1/(cosecθ - cotθ) - 1/(sinθ) is equal to:

Question

Accepted Answer

Step 1: Recall the definitions:
- cosec θ = 1/sin θ
- cot θ = cos θ/sin θ

Step 2: Simplify the denominator of the first term:
  cosec θ - cot θ = (1/sin θ) - (cos θ/sin θ) = (1 - cos θ)/sin θ

Step 3: Therefore,
  1/(cosec θ - cot θ) = 1 / [(1 - cos θ)/sin θ] = sin θ / (1 - cos θ)

Step 4: The expression becomes:
  (sin θ) / (1 - cos θ) - 1 / (sin θ)

Step 5: Find common denominator sin θ (1 - cos θ):
  = [sin² θ - (1 - cos θ)] / [sin θ (1 - cos θ)]

Step 6: Simplify numerator:
  sin² θ - 1 + cos θ = (sin² θ + cos θ - 1)

Step 7: Use identity sin² θ + cos² θ = 1, so sin² θ = 1 - cos² θ:
  (1 - cos² θ) + cos θ - 1 = -cos² θ + cos θ = cos θ - cos² θ

Step 8: Factor numerator:
  cos θ (1 - cos θ)

Step 9: Expression is now:
  [cos θ (1 - cos θ)] / [sin θ (1 - cos θ)]

Step 10: Cancel (1 - cos θ):
  cos θ / sin θ = cot θ

Step 11: Re-check step 4 to 10 for correctness:
  Actually, step 5 numerator should be sin² θ - (1 - cos θ) = sin² θ - 1 + cos θ
  Using sin² θ = 1 - cos² θ:
  (1 - cos² θ) - 1 + cos θ = -cos² θ + cos θ

Step 12: So final expression is cot θ.

But the options say tan θ is correct. Let's verify carefully:

Alternate approach:
1/(cosec θ - cot θ) - 1/sin θ
= 1/[(1/sin θ) - (cos θ/sin θ)] - 1/sin θ
= 1/[(1 - cos θ)/sin θ] - 1/sin θ
= sin θ / (1 - cos θ) - 1/sin θ

Find common denominator sin θ (1 - cos θ):
= [sin² θ - (1 - cos θ)] / [sin θ (1 - cos θ)]
= [sin² θ - 1 + cos θ] / [sin θ (1 - cos θ)]

Using sin² θ = 1 - cos² θ:
= [1 - cos² θ - 1 + cos θ] / [sin θ (1 - cos θ)]
= [-cos² θ + cos θ] / [sin θ (1 - cos θ)]
= cos θ (-cos θ + 1) / [sin θ (1 - cos θ)]
= cos θ (1 - cos θ) / [sin θ (1 - cos θ)]

Cancel (1 - cos θ):
= cos θ / sin θ = cot θ

Therefore, the expression equals cot θ.

Hence, correct answer is cot θ.

Answer

Answer

Answer