Solve frac{5cos^{2}60^{circ}+4cos^{2}30^{circ}-tan^{2}45^{circ}}{cos^260^circ+sin^230^circ}

$$\frac { 5 \cos ^ { 2 } 60 ^ { \circ } + 4 \cos ^ { 2 } 30 ^ { \circ } - \tan ^ { 2 } 45 ^ { \circ } } { \cos ^ { 2 } 60 ^ { \circ } + \sin ^ { 2 } 30 ^ { \circ } }$$

Evaluate

$$\frac{13}{2}=6.5$$

Solution Steps

Get the value of \cos(60) from trigonometric values table.

$$\frac{5\times \left(\frac{1}{2}\right)^{2}+4\left(\cos(30)\right)^{2}-\left(\tan(45)\right)^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{5\times \frac{1}{4}+4\left(\cos(30)\right)^{2}-\left(\tan(45)\right)^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Multiply 5 and \frac{1}{4} to get \frac{5}{4}.

$$\frac{\frac{5}{4}+4\left(\cos(30)\right)^{2}-\left(\tan(45)\right)^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Get the value of \cos(30) from trigonometric values table.

$$\frac{\frac{5}{4}+4\times \left(\frac{\sqrt{3}}{2}\right)^{2}-\left(\tan(45)\right)^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

To raise \frac{\sqrt{3}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\frac{5}{4}+4\times \frac{\left(\sqrt{3}\right)^{2}}{2^{2}}-\left(\tan(45)\right)^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Express 4\times \frac{\left(\sqrt{3}\right)^{2}}{2^{2}} as a single fraction.

$$\frac{\frac{5}{4}+\frac{4\left(\sqrt{3}\right)^{2}}{2^{2}}-\left(\tan(45)\right)^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Expand 2^{2}.

$$\frac{\frac{5}{4}+\frac{4\left(\sqrt{3}\right)^{2}}{4}-\left(\tan(45)\right)^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Since \frac{5}{4} and \frac{4\left(\sqrt{3}\right)^{2}}{4} have the same denominator, add them by adding their numerators.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}}{4}-\left(\tan(45)\right)^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Get the value of \tan(45) from trigonometric values table.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}}{4}-1^{2}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}}{4}-1}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{4}{4}.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}}{4}-\frac{4}{4}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Since \frac{5+4\left(\sqrt{3}\right)^{2}}{4} and \frac{4}{4} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}-4}{4}}{\left(\cos(60)\right)^{2}+\left(\sin(30)\right)^{2}}$$

Get the value of \cos(60) from trigonometric values table.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}-4}{4}}{\left(\frac{1}{2}\right)^{2}+\left(\sin(30)\right)^{2}}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}-4}{4}}{\frac{1}{4}+\left(\sin(30)\right)^{2}}$$

Get the value of \sin(30) from trigonometric values table.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}-4}{4}}{\frac{1}{4}+\left(\frac{1}{2}\right)^{2}}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}-4}{4}}{\frac{1}{4}+\frac{1}{4}}$$

Add \frac{1}{4} and \frac{1}{4} to get \frac{1}{2}.

$$\frac{\frac{5+4\left(\sqrt{3}\right)^{2}-4}{4}}{\frac{1}{2}}$$

Divide \frac{5+4\left(\sqrt{3}\right)^{2}-4}{4} by \frac{1}{2} by multiplying \frac{5+4\left(\sqrt{3}\right)^{2}-4}{4} by the reciprocal of \frac{1}{2}.

$$\frac{\left(5+4\left(\sqrt{3}\right)^{2}-4\right)\times 2}{4}$$

The square of \sqrt{3} is 3.

$$\frac{\left(5+4\times 3-4\right)\times 2}{4}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(5+12-4\right)\times 2}{4}$$

Add 5 and 12 to get 17.

$$\frac{\left(17-4\right)\times 2}{4}$$

Subtract 4 from 17 to get 13.

$$\frac{13\times 2}{4}$$

Multiply 13 and 2 to get 26.

$$\frac{26}{4}$$

Reduce the fraction \frac{26}{4} to lowest terms by extracting and canceling out 2.

$$\frac{13}{2}$$