Solve frac{sin^{2}45^{circ}+cos^245^circ}{tan^260^circ}

$$\frac { \sin ^ { 2 } 45 ^ { \circ } + \cos ^ { 2 } 45 ^ { \circ } } { \tan ^ { 2 } 60 ^ { \circ } }$$

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

Solution Steps

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

$$\frac{\left(\frac{\sqrt{2}}{2}\right)^{2}+\left(\cos(45)\right)^{2}}{\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\cos(45)\right)^{2}}{\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\frac{\sqrt{2}}{2}\right)^{2}}{\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}{\left(\tan(60)\right)^{2}}$$

Combine \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} and \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} to get 2\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}.

$$\frac{2\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}{\left(\tan(60)\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\times \frac{2}{2^{2}}}{\left(\tan(60)\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\times \frac{2}{4}}{\left(\tan(60)\right)^{2}}$$

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

$$\frac{2\times \frac{1}{2}}{\left(\tan(60)\right)^{2}}$$

Multiply 2 and \frac{1}{2} to get 1.

$$\frac{1}{\left(\tan(60)\right)^{2}}$$

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

$$\frac{1}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{1}{3}$$