Solve frac{tan^{2}45^{circ}+cos^{2}60^{circ}}{cos30^{circ}-4cos^245^circ-tan^260^circ}

$$\frac { \tan ^ { 2 } 45 ^ { \circ } + \cos ^ { 2 } 60 ^ { \circ } } { \cos 30 ^ { \circ } - 4 \cos ^ { 2 } 45 ^ { \circ } - \tan ^ { 2 } 60 ^ { \circ } }$$

Evaluate

$$-\frac{5\sqrt{3}}{194}-\frac{25}{97}\approx -0.302372443$$

Short Solution Steps

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

$$\frac{1^{2}+\left(\cos(60)\right)^{2}}{\cos(30)-4\left(\cos(45)\right)^{2}-\left(\tan(60)\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{1+\left(\cos(60)\right)^{2}}{\cos(30)-4\left(\cos(45)\right)^{2}-\left(\tan(60)\right)^{2}}$$

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

$$\frac{1+\left(\frac{1}{2}\right)^{2}}{\cos(30)-4\left(\cos(45)\right)^{2}-\left(\tan(60)\right)^{2}}$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1+\frac{1}{4}}{\cos(30)-4\left(\cos(45)\right)^{2}-\left(\tan(60)\right)^{2}}$$

Add 1 and \frac{1}{4} to get \frac{5}{4}.

$$\frac{\frac{5}{4}}{\cos(30)-4\left(\cos(45)\right)^{2}-\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-4\left(\cos(45)\right)^{2}-\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-4\times \left(\frac{\sqrt{2}}{2}\right)^{2}-\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-4\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}-\left(\tan(60)\right)^{2}}$$

Express 4\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}-\left(\tan(60)\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-\frac{4\times 2}{2^{2}}-\left(\tan(60)\right)^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-\frac{8}{2^{2}}-\left(\tan(60)\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-\frac{8}{4}-\left(\tan(60)\right)^{2}}$$

Divide 8 by 4 to get 2.

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-2-\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}}{2}-\frac{2\times 2}{2}-\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}-2\times 2}{2}-\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}-4}{2}-\left(\tan(60)\right)^{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}-4}{2}-\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}-4}{2}-3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 3 times \frac{2}{2}.

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}-4}{2}-\frac{3\times 2}{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}-4-3\times 2}{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}-4-6}{2}}$$

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

$$\frac{\frac{5}{4}}{\frac{\sqrt{3}-10}{2}}$$

Divide \frac{5}{4} by \frac{\sqrt{3}-10}{2} by multiplying \frac{5}{4} by the reciprocal of \frac{\sqrt{3}-10}{2}.

$$\frac{5\times 2}{4\left(\sqrt{3}-10\right)}$$

Cancel out 2 in both numerator and denominator.

$$\frac{5}{2\left(\sqrt{3}-10\right)}$$

Use the distributive property to multiply 2 by \sqrt{3}-10.

$$\frac{5}{2\sqrt{3}-20}$$

Rationalize the denominator of \frac{5}{2\sqrt{3}-20} by multiplying numerator and denominator by 2\sqrt{3}+20.

$$\frac{5\left(2\sqrt{3}+20\right)}{\left(2\sqrt{3}-20\right)\left(2\sqrt{3}+20\right)}$$

Consider \left(2\sqrt{3}-20\right)\left(2\sqrt{3}+20\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{5\left(2\sqrt{3}+20\right)}{\left(2\sqrt{3}\right)^{2}-20^{2}}$$

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

$$\frac{5\left(2\sqrt{3}+20\right)}{2^{2}\left(\sqrt{3}\right)^{2}-20^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5\left(2\sqrt{3}+20\right)}{4\left(\sqrt{3}\right)^{2}-20^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{5\left(2\sqrt{3}+20\right)}{4\times 3-20^{2}}$$

Multiply 4 and 3 to get 12.

$$\frac{5\left(2\sqrt{3}+20\right)}{12-20^{2}}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{5\left(2\sqrt{3}+20\right)}{12-400}$$

Subtract 400 from 12 to get -388.

$$\frac{5\left(2\sqrt{3}+20\right)}{-388}$$

Use the distributive property to multiply 5 by 2\sqrt{3}+20.

$$\frac{10\sqrt{3}+100}{-388}$$