Question

The value of $\left(\frac{1}{2} \tan^2 45^\circ - \cos^2 60^\circ\right)$ is :

• A. $0$
• B. $-\frac{1}{2}$
• C. $\frac{1}{4}$
• D. $-\frac{1}{4}$

Hint:

Double check your calculations on simple fraction arithmetic.
A common error is confusing $\cos(60^\circ)$ with $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, which would lead to an incorrect answer.
Keeping a clear mental checklist of basic trigonometric values ensures perfect accuracy.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
This question is a straightforward evaluation from "Introduction to Trigonometry".
We are given a numerical expression containing squared trigonometric functions of standard angles ($45^\circ$ and $60^\circ$).
We need to substitute the standard values of these trigonometric ratios and evaluate the arithmetic expression.

Step 2: Key Formula or Approach:
We need the standard trigonometric values for the given angles:

• $\tan(45^\circ) = 1$

• $\cos(60^\circ) = \frac{1}{2}$
We will square these values as indicated in the expression and perform the arithmetic operations.

Step 3: Detailed Explanation:

• Write down the expression to be evaluated:
\[ E = \frac{1}{2} \tan^2 45^\circ - \cos^2 60^\circ \]

• Substitute the standard values $\tan 45^\circ = 1$ and $\cos 60^\circ = \frac{1}{2}$ into the expression:
\[ E = \frac{1}{2} (1)^2 - \left(\frac{1}{2}\right)^2 \]

• Compute the squared terms:
\[ (1)^2 = 1 \]
\[ \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

• Substitute these values back into the expression:
\[ E = \frac{1}{2} (1) - \frac{1}{4} \] \[ E = \frac{1}{2} - \frac{1}{4} \]

• To subtract these fractions, find a common denominator, which is 4:
\[ E = \frac{2}{4} - \frac{1}{4} \] \[ E = \frac{1}{4} \]

Step 4: Final Answer:
The evaluated value of the expression is $\frac{1}{4}$, which corresponds to Option (C).