Solve frac{1-sin^{2}45^{circ}}{1+sin^245^circ}+tan^245^circ

$$\frac { 1 - \sin ^ { 2 } 45 ^ { \circ } } { 1 + \sin ^ { 2 } 45 ^ { \circ } } + \tan ^ { 2 } 45 ^ { \circ }$$

Evaluate

$$\frac{4}{3}\approx 1.333333333$$

Solution Steps

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

$$\frac{1-\left(\frac{\sqrt{2}}{2}\right)^{2}}{1+\left(\sin(45)\right)^{2}}+\left(\tan(45)\right)^{2}$$

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

$$\frac{1-\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}{1+\left(\sin(45)\right)^{2}}+\left(\tan(45)\right)^{2}$$

The square of \sqrt{2} is 2.

$$\frac{1-\frac{2}{2^{2}}}{1+\left(\sin(45)\right)^{2}}+\left(\tan(45)\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{1-\frac{2}{4}}{1+\left(\sin(45)\right)^{2}}+\left(\tan(45)\right)^{2}$$

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

$$\frac{1-\frac{1}{2}}{1+\left(\sin(45)\right)^{2}}+\left(\tan(45)\right)^{2}$$

Subtract \frac{1}{2} from 1 to get \frac{1}{2}.

$$\frac{\frac{1}{2}}{1+\left(\sin(45)\right)^{2}}+\left(\tan(45)\right)^{2}$$

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

$$\frac{\frac{1}{2}}{1+\left(\frac{\sqrt{2}}{2}\right)^{2}}+\left(\tan(45)\right)^{2}$$

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

$$\frac{\frac{1}{2}}{1+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}+\left(\tan(45)\right)^{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2^{2}}{2^{2}}.

$$\frac{\frac{1}{2}}{\frac{2^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}+\left(\tan(45)\right)^{2}$$

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

$$\frac{\frac{1}{2}}{\frac{2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}}}+\left(\tan(45)\right)^{2}$$

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

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

Cancel out 2 in both numerator and denominator.

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

$$\frac{2}{2+4}+\left(\tan(45)\right)^{2}$$

Add 2 and 4 to get 6.

$$\frac{2}{6}+\left(\tan(45)\right)^{2}$$

Reduce the fraction \frac{2}{6} to lowest terms by extracting and canceling out 2.

$$\frac{1}{3}+\left(\tan(45)\right)^{2}$$

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

$$\frac{1}{3}+1^{2}$$

Calculate 1 to the power of 2 and get 1.

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

Add \frac{1}{3} and 1 to get \frac{4}{3}.

$$\frac{4}{3}$$