Solve cos60^circ/1+sin60^circ+1/tan30^circ

$$\frac { \cos 60 ^ { \circ } } { 1 + \sin 60 ^ { \circ } } + \frac { 1 } { \tan 30 ^ { \circ } }$$

Evaluate

$$2$$

Short Solution Steps

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

$$\frac{\frac{1}{2}}{1+\sin(60)}+\frac{1}{\tan(30)}$$

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

$$\frac{\frac{1}{2}}{1+\frac{\sqrt{3}}{2}}+\frac{1}{\tan(30)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

$$\frac{\frac{1}{2}}{\frac{2}{2}+\frac{\sqrt{3}}{2}}+\frac{1}{\tan(30)}$$

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

$$\frac{\frac{1}{2}}{\frac{2+\sqrt{3}}{2}}+\frac{1}{\tan(30)}$$

Divide \frac{1}{2} by \frac{2+\sqrt{3}}{2} by multiplying \frac{1}{2} by the reciprocal of \frac{2+\sqrt{3}}{2}.

$$\frac{2}{2\left(2+\sqrt{3}\right)}+\frac{1}{\tan(30)}$$

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

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

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

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

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

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

The square of \sqrt{3} is 3.

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

Cancel out 3 and 3.

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

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

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

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

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

Do the multiplications in 2+\sqrt{3}\times 2\left(2+\sqrt{3}\right).

$$\frac{2+4\sqrt{3}+6}{2\left(2+\sqrt{3}\right)}$$

Do the calculations in 2+4\sqrt{3}+6.

$$\frac{8+4\sqrt{3}}{2\left(2+\sqrt{3}\right)}$$

Expand 2\left(2+\sqrt{3}\right).

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

Rationalize the denominator of \frac{8+4\sqrt{3}}{2\sqrt{3}+4} by multiplying numerator and denominator by 2\sqrt{3}-4.

$$\frac{\left(8+4\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{\left(2\sqrt{3}+4\right)\left(2\sqrt{3}-4\right)}$$

Consider \left(2\sqrt{3}+4\right)\left(2\sqrt{3}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(8+4\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{\left(2\sqrt{3}\right)^{2}-4^{2}}$$

Expand \left(2\sqrt{3}\right)^{2}.

$$\frac{\left(8+4\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{2^{2}\left(\sqrt{3}\right)^{2}-4^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(8+4\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{4\left(\sqrt{3}\right)^{2}-4^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(8+4\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{4\times 3-4^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{\left(8+4\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{12-4^{2}}$$

Calculate 4 to the power of 2 and get 16.

$$\frac{\left(8+4\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{12-16}$$

Subtract 16 from 12 to get -4.

$$\frac{\left(8+4\sqrt{3}\right)\left(2\sqrt{3}-4\right)}{-4}$$

Use the distributive property to multiply 8+4\sqrt{3} by 2\sqrt{3}-4 and combine like terms.

$$\frac{-32+8\left(\sqrt{3}\right)^{2}}{-4}$$

The square of \sqrt{3} is 3.

$$\frac{-32+8\times 3}{-4}$$

Multiply 8 and 3 to get 24.

$$\frac{-32+24}{-4}$$

Add -32 and 24 to get -8.

$$\frac{-8}{-4}$$

Divide -8 by -4 to get 2.

$$2$$