Solve (frac{tan30^{circ}}{tan60^{circ}})^2-2(sin30^circ/cos60^circ)^2

$$( \frac { \tan 30 ^ { \circ } } { \tan 60 ^ { \circ } } ) ^ { 2 } - 2 ( \frac { \sin 30 ^ { \circ } } { \cos 60 ^ { \circ } } ) ^ { 2 }$$

Evaluate

$$-\frac{17}{9}\approx -1.888888889$$

Solution Steps

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

$$\left(\frac{\frac{\sqrt{3}}{3}}{\tan(60)}\right)^{2}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

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

$$\left(\frac{\frac{\sqrt{3}}{3}}{\sqrt{3}}\right)^{2}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

Express \frac{\frac{\sqrt{3}}{3}}{\sqrt{3}} as a single fraction.

$$\left(\frac{\sqrt{3}}{3\sqrt{3}}\right)^{2}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

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

$$\left(\frac{\sqrt{3}\sqrt{3}}{3\left(\sqrt{3}\right)^{2}}\right)^{2}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

The square of \sqrt{3} is 3.

$$\left(\frac{\sqrt{3}\sqrt{3}}{3\times 3}\right)^{2}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\left(\frac{3}{3\times 3}\right)^{2}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

Multiply 3 and 3 to get 9.

$$\left(\frac{3}{9}\right)^{2}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

Reduce the fraction \frac{3}{9} to lowest terms by extracting and canceling out 3.

$$\left(\frac{1}{3}\right)^{2}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\frac{1}{9}-2\times \left(\frac{\sin(30)}{\cos(60)}\right)^{2}$$

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

$$\frac{1}{9}-2\times \left(\frac{\frac{1}{2}}{\cos(60)}\right)^{2}$$

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

$$\frac{1}{9}-2\times \left(\frac{\frac{1}{2}}{\frac{1}{2}}\right)^{2}$$

Divide \frac{1}{2} by \frac{1}{2} to get 1.

$$\frac{1}{9}-2\times 1^{2}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{1}{9}-2\times 1$$

Multiply 2 and 1 to get 2.

$$\frac{1}{9}-2$$

Subtract 2 from \frac{1}{9} to get -\frac{17}{9}.

$$-\frac{17}{9}$$