Question

Which of the following values is equal to 1 ?

• A. \(\sin^2 60^\circ + \cos 60^\circ\)
• B. \(\sin 90^\circ \times \cos 90^\circ\)
• C. \(\sin^2 60^\circ\)
• D. \(\sin 45^\circ \times \frac{1}{\cos 45^\circ}\)

Hint:

Recognizing identities can save time. The expression in option (D) is equivalent to \(\tan 45^\circ\). Knowing that \(\tan 45^\circ = 1\) allows you to find the answer without calculating with fractions.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question requires evaluating four trigonometric expressions to determine which one equals 1. This involves knowing standard trigonometric values and identities.

Step 2: Key Formula or Approach:
We will use the standard values of trigonometric functions for angles 45°, 60°, and 90°. Key values: \(\sin 60^\circ = \frac{\sqrt{3}}{2}\), \(\cos 60^\circ = \frac{1}{2}\), \(\sin 90^\circ = 1\), \(\cos 90^\circ = 0\), \(\sin 45^\circ = \frac{1}{\sqrt{2}}\), \(\cos 45^\circ = \frac{1}{\sqrt{2}}\). Key identity: \(\frac{\sin \theta}{\cos \theta} = \tan \theta\).

Step 3: Detailed Explanation:
Let's evaluate each option:
(A) \(\sin^2 60^\circ + \cos 60^\circ = (\frac{\sqrt{3}}{2})^2 + \frac{1}{2} = \frac{3}{4} + \frac{1}{2} = \frac{3}{4} + \frac{2}{4} = \frac{5}{4}\). This is not equal to 1.
(B) \(\sin 90^\circ \times \cos 90^\circ = 1 \times 0 = 0\). This is not equal to 1.
(C) \(\sin^2 60^\circ = (\frac{\sqrt{3}}{2})^2 = \frac{3}{4}\). This is not equal to 1.
(D) \(\sin 45^\circ \times \frac{1}{\cos 45^\circ} = \frac{\sin 45^\circ}{\cos 45^\circ} = \tan 45^\circ\). Since \(\tan 45^\circ = 1\), this expression is equal to 1.

Step 4: Final Answer:
The expression that is equal to 1 is \(\sin 45^\circ \times \frac{1}{\cos 45^\circ}\).