Solve frac{cos^{2}20^{circ}+cos^{2}70^{circ}}{2[sin^259^circ+sin^231^circ]}=2/k

$$\frac { \cos ^ { 2 } 20 ^ { \circ } + \cos ^ { 2 } 70 ^ { \circ } } { 2 [ \sin ^ { 2 } 59 ^ { \circ } + \sin ^ { 2 } 31 ^ { \circ } ] } = \frac { 2 } { k }$$

Solve for k

$$k = \frac{24999999999999995991762544708162}{6250000000000000473556281128025} = 3\frac{6.249999999999996 \times 10^{30}}{6.25 \times 10^{30}} \approx 4$$

Steps for Solving Linear Equation

Evaluate trigonometric functions in the problem

$$\frac{0.9396926207859084 ^ {2} + 0.3420201433256688 ^ {2}}{2 {(0.8571673007021122 ^ {2} + 0.5150380749100542 ^ {2})}} = \frac{2}{k}$$

Variable k cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by k.

$$\frac{6250000000000000000000000000000}{12499999999999997995881272354081}k\left(0.9396926207859084^{2}+0.3420201433256688^{2}\right)=2$$

Calculate 0.9396926207859084 to the power of 2 and get 0.88302222155948904756966801319056.

$$\frac{6250000000000000000000000000000}{12499999999999997995881272354081}k\left(0.88302222155948904756966801319056+0.3420201433256688^{2}\right)=2$$

Calculate 0.3420201433256688 to the power of 2 and get 0.11697777844051102819933696729344.

$$\frac{6250000000000000000000000000000}{12499999999999997995881272354081}k\left(0.88302222155948904756966801319056+0.11697777844051102819933696729344\right)=2$$

Add 0.88302222155948904756966801319056 and 0.11697777844051102819933696729344 to get 1.000000000000000075769004980484.

$$\frac{6250000000000000000000000000000}{12499999999999997995881272354081}k\times 1.000000000000000075769004980484=2$$

Convert decimal number 1.000000000000000075769004980484 to fraction \frac{250000000000000018942251245121}{10000000000}. Reduce the fraction \frac{250000000000000018942251245121}{10000000000} to lowest terms by extracting and canceling out 1.

$$\frac{6250000000000000000000000000000}{12499999999999997995881272354081}k\times \frac{250000000000000018942251245121}{250000000000000000000000000000}=2$$

Multiply \frac{6250000000000000000000000000000}{12499999999999997995881272354081} times \frac{250000000000000018942251245121}{250000000000000000000000000000} by multiplying numerator times numerator and denominator times denominator.

$$\frac{6250000000000000000000000000000\times 250000000000000018942251245121}{12499999999999997995881272354081\times 250000000000000000000000000000}k=2$$

Do the multiplications in the fraction \frac{6250000000000000000000000000000\times 250000000000000018942251245121}{12499999999999997995881272354081\times 250000000000000000000000000000}.

$$\frac{1562500000000000118389070282006250000000000000000000000000000}{3124999999999999498970318088520250000000000000000000000000000}k=2$$

Reduce the fraction \frac{1562500000000000118389070282006250000000000000000000000000000}{3124999999999999498970318088520250000000000000000000000000000} to lowest terms by extracting and canceling out 250000000000000000000000000000.

$$\frac{6250000000000000473556281128025}{12499999999999997995881272354081}k=2$$

Multiply both sides by \frac{12499999999999997995881272354081}{6250000000000000473556281128025}, the reciprocal of \frac{6250000000000000473556281128025}{12499999999999997995881272354081}.

$$k=2\times \frac{12499999999999997995881272354081}{6250000000000000473556281128025}$$

Express 2\times \frac{12499999999999997995881272354081}{6250000000000000473556281128025} as a single fraction.

$$k=\frac{2\times 12499999999999997995881272354081}{6250000000000000473556281128025}$$

Multiply 2 and 12499999999999997995881272354081 to get 24999999999999995991762544708162.

$$k=\frac{24999999999999995991762544708162}{6250000000000000473556281128025}$$