What is the exact numerical value of the trigonometric expression \frac{4\cos^2 30^\circ +… — JAMB UTME

What is the exact numerical value of the trigonometric expression $\frac{4\cos^2 30^\circ + 2\sin^2 45^\circ}{\tan^2 60^\circ - \csc 30^\circ}$ ?

Answer: 4

By evaluating the special angles directly:

4\cos^2 30^\circ = 3

2\sin^2 45^\circ = 1

\tan^2 60^\circ = 3

, and

\csc 30^\circ = 2

. Substituting these into the expression yields

\frac{3 + 1}{3 - 2} = \frac{4}{1} = 4

Substitute the exact value of

\cos 30^\circ

4\cos^2 30^\circ = 4 \left(\frac{\sqrt{3}}{2}\right)^2 = 4 \left(\frac{3}{4}\right) = 3

The exact value of

\cos 30^\circ

\frac{\sqrt{3}}{2}

Substitute the exact value of

\sin 45^\circ

2\sin^2 45^\circ = 2 \left(\frac{1}{\sqrt{2}}\right)^2 = 2 \left(\frac{1}{2}\right) = 1

The exact value of

\sin 45^\circ

\frac{1}{\sqrt{2}}

Substitute the exact values of

\tan 60^\circ

and

\csc 30^\circ

\tan^2 60^\circ = (\sqrt{3})^2 = 3

and

\csc 30^\circ = \frac{1}{\sin 30^\circ} = 2

The exact value of

\tan 60^\circ

\sqrt{3}

and

\csc 30^\circ

is the reciprocal of

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

Simplify the entire fraction

\frac{3 + 1}{3 - 2} = \frac{4}{1} = 4

Dividing the simplified numerator (4) by the simplified denominator (1) gives 4.

Evaluation of Special Angle Trigonometric Ratios and Reciprocal Functions

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