Question

The value of \( (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1) \) is:

• A. \( 1 \)
• B. \( 0 \)
• C. \( 3/2 \)
• D. \( 2/3 \)

Hint:

To simplify trigonometric expressions, remember identities such as \( \sin(90^\circ - x) = \cos x \) and use them to reduce the complexity of the expression.

Answer 1

The Correct Option is A

We are given: \[ (\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1). \] We know that: \[ \cot 10^\circ \cot 70^\circ = \frac{\cos 10^\circ}{\sin 10^\circ} \times \frac{\cos 70^\circ}{\sin 70^\circ}. \] Since \( \cos 70^\circ = \sin 10^\circ \), the above expression simplifies to: \[ \cot 10^\circ \cot 70^\circ = 1. \] Thus: \[ (\sin 70^\circ)(1 - 1) = 0. \] Therefore, the value of the expression is \( 1 \).