Solve cos^{2}45^{circ}-1/sin30^{circ}+1/tan30^{circ}+cos^230^circ+sin^245^circ

$$\cos ^ { 2 } 45 ^ { \circ } - \frac { 1 } { \sin 30 ^ { \circ } } + \frac { 1 } { \tan 30 ^ { \circ } } + \cos ^ { 2 } 30 ^ { \circ } + \sin ^ { 2 } 45 ^ { \circ }$$

Evaluate

$$\sqrt{3}-\frac{1}{4}\approx 1.482050808$$

Solution Steps

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

$$\left(\frac{\sqrt{2}}{2}\right)^{2}-\frac{1}{\sin(30)}+\frac{1}{\tan(30)}+\left(\cos(30)\right)^{2}+\left(\sin(45)\right)^{2}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{\sin(30)}+\frac{1}{\tan(30)}+\left(\cos(30)\right)^{2}+\left(\sin(45)\right)^{2}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\frac{1}{\frac{1}{2}}+\frac{1}{\tan(30)}+\left(\cos(30)\right)^{2}+\left(\sin(45)\right)^{2}$$

Divide 1 by \frac{1}{2} by multiplying 1 by the reciprocal of \frac{1}{2}.

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

Multiply 1 and 2 to get 2.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-2+\frac{1}{\tan(30)}+\left(\cos(30)\right)^{2}+\left(\sin(45)\right)^{2}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-2+\frac{1}{\frac{\sqrt{3}}{3}}+\left(\cos(30)\right)^{2}+\left(\sin(45)\right)^{2}$$

Divide 1 by \frac{\sqrt{3}}{3} by multiplying 1 by the reciprocal of \frac{\sqrt{3}}{3}.

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

Rationalize the denominator of \frac{3}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-2+\sqrt{3}+\left(\frac{\sqrt{3}}{2}\right)^{2}+\left(\sin(45)\right)^{2}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-2+\sqrt{3}+\frac{\left(\sqrt{3}\right)^{2}}{2^{2}}+\left(\sin(45)\right)^{2}$$

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

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

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

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

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}}.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply -2+\sqrt{3} times \frac{2^{2}}{2^{2}}.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{3} times \frac{2^{2}}{2^{2}}.

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

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

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

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

Subtract 2 from 1 to get -1.

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

The square of \sqrt{3} is 3.

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

Calculate 2 to the power of 2 and get 4.

$$-1+\sqrt{3}+\frac{3}{4}$$

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

$$-\frac{1}{4}+\sqrt{3}$$