Solve frac{tan63^{circ}+cot23^{circ}}{tan27^circ+cot67^circ}-tan63^circtan67^circ

$$\frac { \tan 63 ^ { \circ } + \cot 23 ^ { \circ } } { \tan 27 ^ { \circ } + \cot 67 ^ { \circ } } - \tan 63 ^ { \circ } \tan 67 ^ { \circ }$$

Evaluate

$$\frac{962310860074706525646095667393}{729687707581276328125000000000000000000000000}\approx 1.318798234 \cdot 10^{-15}$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{1.9626105055051504 + 2.355852365823753}{0.5095254494944288 + 0.4244748162096049} - 1.9626105055051504 \cdot 2.355852365823752$$

Add 1.9626105055051504 and 2.355852365823753 to get 4.3184628713289034.

$$\frac{4.3184628713289034}{0.5095254494944288+0.4244748162096049}-1.9626105055051504\times 2.355852365823752$$

Add 0.5095254494944288 and 0.4244748162096049 to get 0.9340002657040337.

$$\frac{4.3184628713289034}{0.9340002657040337}-1.9626105055051504\times 2.355852365823752$$

Expand \frac{4.3184628713289034}{0.9340002657040337} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{43184628713289034}{9340002657040337}-1.9626105055051504\times 2.355852365823752$$

Multiply 1.9626105055051504 and 2.355852365823752 to get 4.6236206025848584186515746523008.

$$\frac{43184628713289034}{9340002657040337}-4.6236206025848584186515746523008$$

Convert decimal number 4.6236206025848584186515746523008 to fraction \frac{361220359576942063957154269711}{10000000000}. Reduce the fraction \frac{361220359576942063957154269711}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{43184628713289034}{9340002657040337}-\frac{361220359576942063957154269711}{78125000000000000000000000000}$$

Least common multiple of 9340002657040337 and 78125000000000000000000000000 is 729687707581276328125000000000000000000000000. Convert \frac{43184628713289034}{9340002657040337} and \frac{361220359576942063957154269711}{78125000000000000000000000000} to fractions with denominator 729687707581276328125000000000000000000000000.

$$\frac{3373799118225705781250000000000000000000000000}{729687707581276328125000000000000000000000000}-\frac{3373799118225704818939139925293474353904332607}{729687707581276328125000000000000000000000000}$$

Since \frac{3373799118225705781250000000000000000000000000}{729687707581276328125000000000000000000000000} and \frac{3373799118225704818939139925293474353904332607}{729687707581276328125000000000000000000000000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{3373799118225705781250000000000000000000000000-3373799118225704818939139925293474353904332607}{729687707581276328125000000000000000000000000}$$

Subtract 3373799118225704818939139925293474353904332607 from 3373799118225705781250000000000000000000000000 to get 962310860074706525646095667393.

$$\frac{962310860074706525646095667393}{729687707581276328125000000000000000000000000}$$

Factor

$$\frac{13 \cdot 13369201 \cdot 995146553 \cdot 206070476231 \cdot 3 ^ {3}}{15512821 \cdot 602082797 \cdot 2 ^ {24} \cdot 5 ^ {31}} = 1.3187982339246405 \times 10^{-15}$$