Solve 1+cos^245^circ/2sin45^circ

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

Evaluate

$$\frac{3\sqrt{2}}{4}\approx 1.060660172$$

Solution Steps

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

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

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

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

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

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

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.

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

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

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

Cancel out 2 and 2.

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

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

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

Rationalize the denominator of \frac{2^{2}+\left(\sqrt{2}\right)^{2}}{2^{2}\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Add 4 and 2 to get 6.

$$\frac{6\sqrt{2}}{2^{2}\times 2}$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\frac{6\sqrt{2}}{2^{3}}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{6\sqrt{2}}{8}$$

Divide 6\sqrt{2} by 8 to get \frac{3}{4}\sqrt{2}.

$$\frac{3}{4}\sqrt{2}$$