Solve frac{(sec^{2}27^{circ}-cot^{2}63^{circ})+(sin^{2}52^{circ}+sin^{2}38^{circ})}{(csc^{2}34^{circ}-tan^{2}56^{circ})+tan10^{circ}tan20^circ*tan30^circ*tan70^circtan80^circ}

$$\frac { ( \sec ^ { 2 } 27 ^ { \circ } - \cot ^ { 2 } 63 ^ { \circ } ) + ( \sin ^ { 2 } 52 ^ { \circ } + \sin ^ { 2 } 38 ^ { \circ } ) } { ( \csc ^ { 2 } 34 ^ { \circ } - \tan ^ { 2 } 56 ^ { \circ } ) + \tan 10 ^ { \circ } \tan 20 ^ { \circ } \cdot \tan 30 ^ { \circ } \cdot \tan 70 ^ { \circ } \tan 80 ^ { \circ } }$$

Evaluate

$$\frac{999999999999999931448803767966335000000000000000}{215729094401737544005222019593172236821986564723}\approx 4.635443368$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{{(1.1223262376343608 ^ {2} - 0.5095254494944289 ^ {2})} + {(0.788010753606722 ^ {2} + 0.6156614753256582 ^ {2})}}{{(1.7882916499714003 ^ {2} - 1.4825609685127403 ^ {2})} + 0.17632698070846498 \cdot -0.5685418111965239 \cdot 5.671281819617707}$$

Calculate 1.1223262376343608 to the power of 2 and get 1.25961618368249970853103282457664.

$$\frac{1.25961618368249970853103282457664-0.5095254494944289^{2}+0.788010753606722^{2}+0.6156614753256582^{2}}{1.7882916499714003^{2}-1.4825609685127403^{2}+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Calculate 0.5095254494944289 to the power of 2 and get 0.25961618368249981578661213715521.

$$\frac{1.25961618368249970853103282457664-0.25961618368249981578661213715521+0.788010753606722^{2}+0.6156614753256582^{2}}{1.7882916499714003^{2}-1.4825609685127403^{2}+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Subtract 0.25961618368249981578661213715521 from 1.25961618368249970853103282457664 to get 0.99999999999999989274442068742143.

$$\frac{0.99999999999999989274442068742143+0.788010753606722^{2}+0.6156614753256582^{2}}{1.7882916499714003^{2}-1.4825609685127403^{2}+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Calculate 0.788010753606722 to the power of 2 and get 0.620960947799833929531443585284.

$$\frac{0.99999999999999989274442068742143+0.620960947799833929531443585284+0.6156614753256582^{2}}{1.7882916499714003^{2}-1.4825609685127403^{2}+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Add 0.99999999999999989274442068742143 and 0.620960947799833929531443585284 to get 1.62096094779983382227586427270543.

$$\frac{1.62096094779983382227586427270543+0.6156614753256582^{2}}{1.7882916499714003^{2}-1.4825609685127403^{2}+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Calculate 0.6156614753256582 to the power of 2 and get 0.37903905220016604062174326322724.

$$\frac{1.62096094779983382227586427270543+0.37903905220016604062174326322724}{1.7882916499714003^{2}-1.4825609685127403^{2}+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Add 1.62096094779983382227586427270543 and 0.37903905220016604062174326322724 to get 1.99999999999999986289760753593267.

$$\frac{1.99999999999999986289760753593267}{1.7882916499714003^{2}-1.4825609685127403^{2}+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Calculate 1.7882916499714003 to the power of 2 and get 3.19798702535743329059580794284009.

$$\frac{1.99999999999999986289760753593267}{3.19798702535743329059580794284009-1.4825609685127403^{2}+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Calculate 1.4825609685127403 to the power of 2 and get 2.19798702535743453526412341524409.

$$\frac{1.99999999999999986289760753593267}{3.19798702535743329059580794284009-2.19798702535743453526412341524409+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Subtract 2.19798702535743453526412341524409 from 3.19798702535743329059580794284009 to get 0.999999999999998755331684527596.

$$\frac{1.99999999999999986289760753593267}{0.999999999999998755331684527596+0.17632698070846498\left(-0.5685418111965239\right)\times 5.671281819617707}$$

Multiply 0.17632698070846498 and -0.5685418111965239 to get -0.100249260974805208683331080883022.

$$\frac{1.99999999999999986289760753593267}{0.999999999999998755331684527596-0.100249260974805208683331080883022\times 5.671281819617707}$$

Multiply -0.100249260974805208683331080883022 and 5.671281819617707 to get -0.568541811196523667321240488409655526356026870554.

$$\frac{1.99999999999999986289760753593267}{0.999999999999998755331684527596-0.568541811196523667321240488409655526356026870554}$$

Subtract 0.568541811196523667321240488409655526356026870554 from 0.999999999999998755331684527596 to get 0.431458188803475088010444039186344473643973129446.

$$\frac{1.99999999999999986289760753593267}{0.431458188803475088010444039186344473643973129446}$$

Expand \frac{1.99999999999999986289760753593267}{0.431458188803475088010444039186344473643973129446} by multiplying both numerator and the denominator by 1000000000000000000000000000000000000000000000000.

$$\frac{1999999999999999862897607535932670000000000000000}{431458188803475088010444039186344473643973129446}$$

Reduce the fraction \frac{1999999999999999862897607535932670000000000000000}{431458188803475088010444039186344473643973129446} to lowest terms by extracting and canceling out 2.

$$\frac{999999999999999931448803767966335000000000000000}{215729094401737544005222019593172236821986564723}$$