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: Work the whole expression out in decimals instead of fractions.
We need the standard values $\tan 45^\circ = 1$ and $\cos 60^\circ = 0.5$.

Step 2: Compute each squared term as a decimal.
\[ \tan^2 45^\circ = (1)^2 = 1 \]
\[ \cos^2 60^\circ = (0.5)^2 = 0.25 \]

Step 3: Substitute into the expression and simplify in decimal form.
\[ \frac{1}{2}\tan^2 45^\circ - \cos^2 60^\circ = \frac{1}{2}(1) - 0.25 = 0.5 - 0.25 = 0.25 \]

Step 4: Convert the decimal answer back to a fraction to match the answer choices.
\[ 0.25 = \frac{25}{100} = \frac{1}{4} \]

Final Answer:
The value of the expression is $\frac{1}{4}$, which matches Option (C). \[ \boxed{\dfrac{1}{4}} \]