Solve left(cos(45)right)^2+left(sin(45)right)^2tan(60)-sqrt[3]{8}

$${ \left( \cos ( 45 ) \right) }^{ 2 } + { \left( \sin ( 45 ) \right) }^{ 2 } \tan ( 60 ) - \sqrt[3]{ 8 }$$

Evaluate

$$\frac{\sqrt{3}-3}{2}\approx -0.633974596$$

Solution Steps

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

$$\left(\frac{\sqrt{2}}{2}\right)^{2}+\left(\sin(45)\right)^{2}\tan(60)-\sqrt[3]{8}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\sin(45)\right)^{2}\tan(60)-\sqrt[3]{8}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\left(\frac{\sqrt{2}}{2}\right)^{2}\tan(60)-\sqrt[3]{8}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}\tan(60)-\sqrt[3]{8}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}\sqrt{3}-\sqrt[3]{8}$$

Express \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}\sqrt{3} as a single fraction.

$$\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}+\frac{\left(\sqrt{2}\right)^{2}\sqrt{3}}{2^{2}}-\sqrt[3]{8}$$

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

$$\frac{\left(\sqrt{2}\right)^{2}+\left(\sqrt{2}\right)^{2}\sqrt{3}}{2^{2}}-\sqrt[3]{8}$$

Calculate \sqrt[3]{8} and get 2.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

The square of \sqrt{2} is 2.

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

Calculate 2 to the power of 2 and get 4.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 2 times \frac{4}{4}.

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

Since \frac{2+2\sqrt{3}}{4} and \frac{2\times 4}{4} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 2+2\sqrt{3}-2\times 4.

$$\frac{2+2\sqrt{3}-8}{4}$$

Do the calculations in 2+2\sqrt{3}-8.

$$\frac{-6+2\sqrt{3}}{4}$$