Solve frac{cos58^{circ}}{sin32^{circ}}+frac{sin22^{circ}}{cos68^{circ}}-frac{cos38^{circ}csc52^{circ}}{tan18^{circ}tan35^circtan60^circtan72^circtan55^circ}

$$\frac { \cos 58 ^ { \circ } } { \sin 32 ^ { \circ } } + \frac { \sin 22 ^ { \circ } } { \cos 68 ^ { \circ } } - \frac { \cos 38 ^ { \circ } \csc 52 ^ { \circ } } { \tan 18 ^ { \circ } \tan 35 ^ { \circ } \tan 60 ^ { \circ } \tan 72 ^ { \circ } \tan 55 ^ { \circ } }$$

Evaluate

$$-\frac{138888888888888894350708549263000000000000000000000000000000000\sqrt{3}}{416666666666666460920589060879894636568477788277176241566052857}+\frac{18730329670795599}{9365164835397799}\approx 1.422649731$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{0.5299192642332049}{0.5299192642332049} + \frac{0.374606593415912}{0.37460659341591196} - \frac{0.788010753606722 \cdot 1.2690182150725788}{0.3249196962329063 \cdot 0.7002075382097097 \tan(60) 3.0776835371752527 \cdot 1.4281480067421144}$$

Divide 0.5299192642332049 by 0.5299192642332049 to get 1.

$$1+\frac{0.374606593415912}{0.37460659341591196}-\frac{0.788010753606722\times 1.2690182150725788}{0.3249196962329063\times 0.7002075382097097\times 1.4281480067421144\times 3.0776835371752527\tan(60)}$$

Expand \frac{0.374606593415912}{0.37460659341591196} by multiplying both numerator and the denominator by 100000000000000000.

$$1+\frac{37460659341591200}{37460659341591196}-\frac{0.788010753606722\times 1.2690182150725788}{0.3249196962329063\times 0.7002075382097097\times 1.4281480067421144\times 3.0776835371752527\tan(60)}$$

Reduce the fraction \frac{37460659341591200}{37460659341591196} to lowest terms by extracting and canceling out 4.

$$1+\frac{9365164835397800}{9365164835397799}-\frac{0.788010753606722\times 1.2690182150725788}{0.3249196962329063\times 0.7002075382097097\times 1.4281480067421144\times 3.0776835371752527\tan(60)}$$

Add 1 and \frac{9365164835397800}{9365164835397799} to get \frac{18730329670795599}{9365164835397799}.

$$\frac{18730329670795599}{9365164835397799}-\frac{0.788010753606722\times 1.2690182150725788}{0.3249196962329063\times 0.7002075382097097\times 1.4281480067421144\times 3.0776835371752527\tan(60)}$$

Multiply 0.788010753606722 and 1.2690182150725788 to get 1.0000000000000000393251015546936.

$$\frac{18730329670795599}{9365164835397799}-\frac{1.0000000000000000393251015546936}{0.3249196962329063\times 0.7002075382097097\times 1.4281480067421144\times 3.0776835371752527\tan(60)}$$

Multiply 0.3249196962329063 and 0.7002075382097097 to get 0.22751122061509000692878331030111.

$$\frac{18730329670795599}{9365164835397799}-\frac{1.0000000000000000393251015546936}{0.22751122061509000692878331030111\times 1.4281480067421144\times 3.0776835371752527\tan(60)}$$

Multiply 0.22751122061509000692878331030111 and 1.4281480067421144 to get 0.324919696232906239885903220050559375140782066984.

$$\frac{18730329670795599}{9365164835397799}-\frac{1.0000000000000000393251015546936}{0.324919696232906239885903220050559375140782066984\times 3.0776835371752527\tan(60)}$$

Multiply 0.324919696232906239885903220050559375140782066984 and 3.0776835371752527 to get 0.9999999999999995062094137461117471277643466918652229797585268568.

$$\frac{18730329670795599}{9365164835397799}-\frac{1.0000000000000000393251015546936}{0.9999999999999995062094137461117471277643466918652229797585268568\tan(60)}$$

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

$$\frac{18730329670795599}{9365164835397799}-\frac{1.0000000000000000393251015546936}{0.9999999999999995062094137461117471277643466918652229797585268568\sqrt{3}}$$

Rationalize the denominator of \frac{1.0000000000000000393251015546936}{0.9999999999999995062094137461117471277643466918652229797585268568\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{18730329670795599}{9365164835397799}-\frac{1.0000000000000000393251015546936\sqrt{3}}{0.9999999999999995062094137461117471277643466918652229797585268568\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{18730329670795599}{9365164835397799}-\frac{1.0000000000000000393251015546936\sqrt{3}}{0.9999999999999995062094137461117471277643466918652229797585268568\times 3}$$

Multiply 0.9999999999999995062094137461117471277643466918652229797585268568 and 3 to get 2.9999999999999985186282412383352413832930400755956689392755805704.

$$\frac{18730329670795599}{9365164835397799}-\frac{1.0000000000000000393251015546936\sqrt{3}}{2.9999999999999985186282412383352413832930400755956689392755805704}$$

Divide 1.0000000000000000393251015546936\sqrt{3} by 2.9999999999999985186282412383352413832930400755956689392755805704 to get \frac{138888888888888894350708549263000000000000000000000000000000000}{416666666666666460920589060879894636568477788277176241566052857}\sqrt{3}.

$$\frac{18730329670795599}{9365164835397799}-\frac{138888888888888894350708549263000000000000000000000000000000000}{416666666666666460920589060879894636568477788277176241566052857}\sqrt{3}$$