Simplify the trigonometric expression \frac{\tan 60^\circ + \sin 45^\circ}{\cos 45^\circ} … — JAMB UTME

Simplify the trigonometric expression $\frac{\tan 60^\circ + \sin 45^\circ}{\cos 45^\circ}$ to its exact surd form.

\sqrt{6} + 1

Substituting the exact values gives

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

\sin 45^\circ = \frac{1}{\sqrt{2}}

, and

\cos 45^\circ = \frac{1}{\sqrt{2}}

. Simplifying

\frac{\sqrt{3} + 1/\sqrt{2}}{1/\sqrt{2}}

gives

\sqrt{3} \cdot \sqrt{2} + 1 = \sqrt{6} + 1

Substitute the exact trigonometric values for the special angles

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

\sin 45^\circ = \frac{1}{\sqrt{2}}

, and

\cos 45^\circ = \frac{1}{\sqrt{2}}

Special angle values must be expressed in exact surd form.

Set up the fractional expression

\frac{\sqrt{3} + \frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}

Replace each ratio with its exact surd equivalent.

Divide numerator terms by the denominator

\frac{\sqrt{3}}{\frac{1}{\sqrt{2}}} + \frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}} = \sqrt{3} \cdot \sqrt{2} + 1 = \sqrt{6} + 1

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Evaluation of Special Angle Trigonometric Ratios and Simplification of Surds

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