Solve frac{1+cos20^{circ}}{2sin20^circ}-sin10^circ(1/tan5^circ-tan5^circ)

$$\frac { 1 + \cos 20 ^ { \circ } } { 2 \sin 20 ^ { \circ } } - \sin 10 ^ { \circ } ( \frac { 1 } { \tan 5 ^ { \circ } } - \tan 5 ^ { \circ } )$$

Evaluate

$$\frac{32800827502616911392781732164613}{17101007166283435000000000000000}\approx 1.918064076$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{1 + 0.9396926207859084}{2 \cdot 0.3420201433256687} - 0.9175768341860202$$

Add 1 and 0.9396926207859084 to get 1.9396926207859084.

$$\frac{1.9396926207859084}{2\times 0.3420201433256687}-0.9175768341860202$$

Multiply 2 and 0.3420201433256687 to get 0.6840402866513374.

$$\frac{1.9396926207859084}{0.6840402866513374}-0.9175768341860202$$

Expand \frac{1.9396926207859084}{0.6840402866513374} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{19396926207859084}{6840402866513374}-0.9175768341860202$$

Reduce the fraction \frac{19396926207859084}{6840402866513374} to lowest terms by extracting and canceling out 2.

$$\frac{9698463103929542}{3420201433256687}-0.9175768341860202$$

Convert decimal number 0.9175768341860202 to fraction \frac{4587884170930101}{10000000000}. Reduce the fraction \frac{4587884170930101}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{9698463103929542}{3420201433256687}-\frac{4587884170930101}{5000000000000000}$$

Least common multiple of 3420201433256687 and 5000000000000000 is 17101007166283435000000000000000. Convert \frac{9698463103929542}{3420201433256687} and \frac{4587884170930101}{5000000000000000} to fractions with denominator 17101007166283435000000000000000.

$$\frac{48492315519647710000000000000000}{17101007166283435000000000000000}-\frac{15691488017030798607218267835387}{17101007166283435000000000000000}$$

Since \frac{48492315519647710000000000000000}{17101007166283435000000000000000} and \frac{15691488017030798607218267835387}{17101007166283435000000000000000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{48492315519647710000000000000000-15691488017030798607218267835387}{17101007166283435000000000000000}$$

Subtract 15691488017030798607218267835387 from 48492315519647710000000000000000 to get 32800827502616911392781732164613.

$$\frac{32800827502616911392781732164613}{17101007166283435000000000000000}$$