Question

Evaluate:

$3 \cos^{2} 60^{\circ} \sec^{2} 30^{\circ} - 2 \sin^{2} 30^{\circ} \tan^{2} 60^{\circ}$

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Answer

To evaluate the given expression $$ 3 \cos^{2} 60^{\circ} \sec^{2} 30^{\circ} - 2 \sin^{2} 30^{\circ} \tan^{2} 60^{\circ} $$ we first need to find the values of the trigonometric functions at these specific angles.

Step 1: Find Trigonometric Values

  1. $\sin 30^\circ = \frac{1}{2}$

  2. $\cos 60^\circ = \frac{1}{2}$

  3. $\tan 60^\circ = \sqrt{3}$

  4. $\sec 30^\circ = \frac{1}{\cos 30^\circ} = 2/\sqrt{3}$ (since $\cos 30^\circ = \sqrt{3}/2$)

Step 2: Plugging in the Values

Substitute these values into the expression, taking care to square them as indicated: $$ 3 \left(\cos 60^\circ\right)^2 \left(\sec 30^\circ\right)^2 - 2 \left(\sin 30^\circ\right)^2 \left(\tan 60^\circ\right)^2 \ = 3 \left(\frac{1}{2}\right)^2 \left(\frac{2}{\sqrt{3}}\right)^2 - 2 \left(\frac{1}{2}\right)^2 \left(\sqrt{3}\right)^2 $$

Step 3: Simplify the Expression

  • For the first term: $$ 3 \left(\frac{1}{4}\right) \left(\frac{4}{3}\right) = 3 \times \frac{1}{3} = 1 $$

  • For the second term: $$ 2 \left(\frac{1}{4}\right) \left(3\right) = \frac{1}{2} \times 3 = \frac{3}{2} $$ Thus, the expression becomes: $$ 1 - \frac{3}{2} = -\frac{1}{2} $$

Final Answer

The evaluated value of the expression is: $$ \boxed{-\frac{1}{2}} $$


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