Solve cos60^{circ}-sin^{2}45^{circ}+1/4tan^230^circ+cos30^circ-sin30^circ

$$\cos 60 ^ { \circ } - \sin ^ { 2 } 45 ^ { \circ } + \frac { 1 } { 4 } \tan ^ { 2 } 30 ^ { \circ } + \cos 30 ^ { \circ } - \sin 30 ^ { \circ }$$

Evaluate

$$\frac{\sqrt{3}}{2}-\frac{5}{12}\approx 0.449358737$$

Short Solution Steps

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

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

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

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

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

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

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

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

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

Subtract \frac{1}{2} from \frac{1}{2} to get 0.

$$0+\frac{1}{4}\left(\tan(30)\right)^{2}+\cos(30)-\sin(30)$$

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

$$0+\frac{1}{4}\times \left(\frac{\sqrt{3}}{3}\right)^{2}+\cos(30)-\sin(30)$$

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

$$0+\frac{1}{4}\times \frac{\left(\sqrt{3}\right)^{2}}{3^{2}}+\cos(30)-\sin(30)$$

Multiply \frac{1}{4} times \frac{\left(\sqrt{3}\right)^{2}}{3^{2}} by multiplying numerator times numerator and denominator times denominator.

$$0+\frac{\left(\sqrt{3}\right)^{2}}{4\times 3^{2}}+\cos(30)-\sin(30)$$

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

$$0+\frac{\left(\sqrt{3}\right)^{2}}{4\times 3^{2}}+\frac{\sqrt{3}}{2}-\sin(30)$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4\times 3^{2} and 2 is 36. Multiply \frac{\sqrt{3}}{2} times \frac{18}{18}.

$$0+\frac{\left(\sqrt{3}\right)^{2}}{36}+\frac{18\sqrt{3}}{36}-\sin(30)$$

Since \frac{\left(\sqrt{3}\right)^{2}}{36} and \frac{18\sqrt{3}}{36} have the same denominator, add them by adding their numerators.

$$0+\frac{\left(\sqrt{3}\right)^{2}+18\sqrt{3}}{36}-\sin(30)$$

Anything plus zero gives itself.

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

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

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

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4\times 3^{2} and 2 is 36. Multiply \frac{\sqrt{3}}{2} times \frac{18}{18}.

$$\frac{\left(\sqrt{3}\right)^{2}}{36}+\frac{18\sqrt{3}}{36}-\frac{1}{2}$$

Since \frac{\left(\sqrt{3}\right)^{2}}{36} and \frac{18\sqrt{3}}{36} have the same denominator, add them by adding their numerators.

$$\frac{\left(\sqrt{3}\right)^{2}+18\sqrt{3}}{36}-\frac{1}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4\times 3^{2} and 2 is 36. Multiply \frac{1}{2} times \frac{18}{18}.

$$\frac{\left(\sqrt{3}\right)^{2}}{36}+\frac{\sqrt{3}}{2}-\frac{18}{36}$$

Since \frac{\left(\sqrt{3}\right)^{2}}{36} and \frac{18}{36} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(\sqrt{3}\right)^{2}-18}{36}+\frac{\sqrt{3}}{2}$$

Since \frac{\sqrt{3}}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

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

The square of \sqrt{3} is 3.

$$\frac{3}{4\times 3^{2}}+\frac{\sqrt{3}-1}{2}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{3}{4\times 9}+\frac{\sqrt{3}-1}{2}$$

Multiply 4 and 9 to get 36.

$$\frac{3}{36}+\frac{\sqrt{3}-1}{2}$$

Reduce the fraction \frac{3}{36} to lowest terms by extracting and canceling out 3.

$$\frac{1}{12}+\frac{\sqrt{3}-1}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 12 and 2 is 12. Multiply \frac{\sqrt{3}-1}{2} times \frac{6}{6}.

$$\frac{1}{12}+\frac{6\left(\sqrt{3}-1\right)}{12}$$

Since \frac{1}{12} and \frac{6\left(\sqrt{3}-1\right)}{12} have the same denominator, add them by adding their numerators.

$$\frac{1+6\left(\sqrt{3}-1\right)}{12}$$

Do the multiplications in 1+6\left(\sqrt{3}-1\right).

$$\frac{1+6\sqrt{3}-6}{12}$$

Do the calculations in 1+6\sqrt{3}-6.

$$\frac{-5+6\sqrt{3}}{12}$$