Solve frac{1-sqrt{3}tan75^circ}{sqrt{3}+tan75^circ}

$$\frac { 1 - \sqrt { 3 } \tan 75 ^ { \circ } } { \sqrt { 3 } + \tan 75 ^ { \circ } }$$

Evaluate

$$\frac{23325317547305485000000000000000-23325317547305486658831356443409\sqrt{3}}{17075317547305486658831356443409}\approx -1$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{1 - \sqrt{3} 3.7320508075688776}{\sqrt{3} + 3.7320508075688776}$$

Rationalize the denominator of \frac{1-3.7320508075688776\sqrt{3}}{\sqrt{3}+3.7320508075688776} by multiplying numerator and denominator by \sqrt{3}-3.7320508075688776.

$$\frac{\left(1-3.7320508075688776\sqrt{3}\right)\left(\sqrt{3}-3.7320508075688776\right)}{\left(\sqrt{3}+3.7320508075688776\right)\left(\sqrt{3}-3.7320508075688776\right)}$$

Consider \left(\sqrt{3}+3.7320508075688776\right)\left(\sqrt{3}-3.7320508075688776\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(1-3.7320508075688776\sqrt{3}\right)\left(\sqrt{3}-3.7320508075688776\right)}{\left(\sqrt{3}\right)^{2}-3.7320508075688776^{2}}$$

Square \sqrt{3}. Square 3.7320508075688776.

$$\frac{\left(1-3.7320508075688776\sqrt{3}\right)\left(\sqrt{3}-3.7320508075688776\right)}{3-13.92820323027551146165206812378176}$$

Subtract 13.92820323027551146165206812378176 from 3 to get -10.92820323027551146165206812378176.

$$\frac{\left(1-3.7320508075688776\sqrt{3}\right)\left(\sqrt{3}-3.7320508075688776\right)}{-10.92820323027551146165206812378176}$$

Apply the distributive property by multiplying each term of 1-3.7320508075688776\sqrt{3} by each term of \sqrt{3}-3.7320508075688776.

$$\frac{\sqrt{3}-3.7320508075688776-3.7320508075688776\left(\sqrt{3}\right)^{2}+13.92820323027551146165206812378176\sqrt{3}}{-10.92820323027551146165206812378176}$$

The square of \sqrt{3} is 3.

$$\frac{\sqrt{3}-3.7320508075688776-3.7320508075688776\times 3+13.92820323027551146165206812378176\sqrt{3}}{-10.92820323027551146165206812378176}$$

Multiply -3.7320508075688776 and 3 to get -11.1961524227066328.

$$\frac{\sqrt{3}-3.7320508075688776-11.1961524227066328+13.92820323027551146165206812378176\sqrt{3}}{-10.92820323027551146165206812378176}$$

Subtract 11.1961524227066328 from -3.7320508075688776 to get -14.9282032302755104.

$$\frac{\sqrt{3}-14.9282032302755104+13.92820323027551146165206812378176\sqrt{3}}{-10.92820323027551146165206812378176}$$

Combine \sqrt{3} and 13.92820323027551146165206812378176\sqrt{3} to get 14.92820323027551146165206812378176\sqrt{3}.

$$\frac{14.92820323027551146165206812378176\sqrt{3}-14.9282032302755104}{-10.92820323027551146165206812378176}$$

Multiply both numerator and denominator by -1.

$$\frac{-14.92820323027551146165206812378176\sqrt{3}+14.9282032302755104}{10.92820323027551146165206812378176}$$