Solve 1-1/cosfrac{pi{3}+1sinpi/3}

$$\frac { 1 - 1 } { \cos \frac { \pi } { 3 } + 1 \sin \frac { \pi } { 3 } }$$

Evaluate

$$0$$

Solution Steps

Subtract 1 from 1 to get 0.

$$\frac{0}{\cos(\frac{\pi }{3})+1\sin(\frac{\pi }{3})}$$

Get the value of \cos(\frac{\pi }{3}) from trigonometric values table.

$$\frac{0}{\frac{1}{2}+1\sin(\frac{\pi }{3})}$$

Get the value of \sin(\frac{\pi }{3}) from trigonometric values table.

$$\frac{0}{\frac{1}{2}+1\times \frac{\sqrt{3}}{2}}$$

Express 1\times \frac{\sqrt{3}}{2} as a single fraction.

$$\frac{0}{\frac{1}{2}+\frac{\sqrt{3}}{2}}$$

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

$$\frac{0}{\frac{1+\sqrt{3}}{2}}$$

Divide 0 by \frac{1+\sqrt{3}}{2} by multiplying 0 by the reciprocal of \frac{1+\sqrt{3}}{2}.

$$\frac{0\times 2}{1+\sqrt{3}}$$

Rationalize the denominator of \frac{0\times 2}{1+\sqrt{3}} by multiplying numerator and denominator by 1-\sqrt{3}.

$$\frac{0\times 2\left(1-\sqrt{3}\right)}{\left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right)}$$

Consider \left(1+\sqrt{3}\right)\left(1-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{0\times 2\left(1-\sqrt{3}\right)}{1^{2}-\left(\sqrt{3}\right)^{2}}$$

Square 1. Square \sqrt{3}.

$$\frac{0\times 2\left(1-\sqrt{3}\right)}{1-3}$$

Subtract 3 from 1 to get -2.

$$\frac{0\times 2\left(1-\sqrt{3}\right)}{-2}$$

Multiply 0 and 2 to get 0.

$$\frac{0\left(1-\sqrt{3}\right)}{-2}$$

Anything times zero gives zero.

$$\frac{0}{-2}$$

Zero divided by any non-zero number gives zero.

$$0$$

Factor

$$0$$