Solve left(sec(54)right)^2-left(cot(36)right)^2/left(csc(57)right)^2-left(tan(33)right)^2+2{left(sin(38{left(sec(52)right)}^2)right)}^2-{left(sin(45)right)}^2

$$\frac{ { \left( \sec ( 54 ) \right) }^{ 2 } - { \left( \cot ( 36 ) \right) }^{ 2 } }{ { \left( \csc ( 57 ) \right) }^{ 2 } - { \left( \tan ( 33 ) \right) }^{ 2 } } +2 { \left( \sin ( 38 { \left( \sec ( 52 ) \right) }^{ 2 } ) \right) }^{ 2 } - { \left( \sin ( 45 ) \right) }^{ 2 }$$

Evaluate

$$\frac{50763106791488245591139219751776422141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}\approx 2.436629126$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{1.7013016167040798 ^ {2} - 1.3763819204711738 ^ {2}}{1.1923632928359476 ^ {2} - 0.6494075931975106 ^ {2}} + 2 \cdot 0.9840297571698318 ^ {2} - {(\sin^{2}(45))}$$

Calculate 1.7013016167040798 to the power of 2 and get 2.89442719099991565956164196476804.

$$\frac{2.89442719099991565956164196476804-1.3763819204711738^{2}}{1.1923632928359476^{2}-0.6494075931975106^{2}}+2\times 0.9840297571698318^{2}-\left(\sin(45)\right)^{2}$$

Calculate 1.3763819204711738 to the power of 2 and get 1.89442719099991659921739674980644.

$$\frac{2.89442719099991565956164196476804-1.89442719099991659921739674980644}{1.1923632928359476^{2}-0.6494075931975106^{2}}+2\times 0.9840297571698318^{2}-\left(\sin(45)\right)^{2}$$

Subtract 1.89442719099991659921739674980644 from 2.89442719099991565956164196476804 to get 0.9999999999999990603442452149616.

$$\frac{0.9999999999999990603442452149616}{1.1923632928359476^{2}-0.6494075931975106^{2}}+2\times 0.9840297571698318^{2}-\left(\sin(45)\right)^{2}$$

Calculate 1.1923632928359476 to the power of 2 and get 1.42173022210258372924980678994576.

$$\frac{0.9999999999999990603442452149616}{1.42173022210258372924980678994576-0.6494075931975106^{2}}+2\times 0.9840297571698318^{2}-\left(\sin(45)\right)^{2}$$

Calculate 0.6494075931975106 to the power of 2 and get 0.42173022210258341571498203711236.

$$\frac{0.9999999999999990603442452149616}{1.42173022210258372924980678994576-0.42173022210258341571498203711236}+2\times 0.9840297571698318^{2}-\left(\sin(45)\right)^{2}$$

Subtract 0.42173022210258341571498203711236 from 1.42173022210258372924980678994576 to get 1.0000000000000003135348247528334.

$$\frac{0.9999999999999990603442452149616}{1.0000000000000003135348247528334}+2\times 0.9840297571698318^{2}-\left(\sin(45)\right)^{2}$$

Expand \frac{0.9999999999999990603442452149616}{1.0000000000000003135348247528334} by multiplying both numerator and the denominator by 10000000000000000000000000000000.

$$\frac{9999999999999990603442452149616}{10000000000000003135348247528334}+2\times 0.9840297571698318^{2}-\left(\sin(45)\right)^{2}$$

Reduce the fraction \frac{9999999999999990603442452149616}{10000000000000003135348247528334} to lowest terms by extracting and canceling out 6.

$$\frac{1666666666666665100573742024936}{1666666666666667189224707921389}+2\times 0.9840297571698318^{2}-\left(\sin(45)\right)^{2}$$

Calculate 0.9840297571698318 to the power of 2 and get 0.96831456299571813879858804029124.

$$\frac{1666666666666665100573742024936}{1666666666666667189224707921389}+2\times 0.96831456299571813879858804029124-\left(\sin(45)\right)^{2}$$

Multiply 2 and 0.96831456299571813879858804029124 to get 1.93662912599143627759717608058248.

$$\frac{1666666666666665100573742024936}{1666666666666667189224707921389}+1.93662912599143627759717608058248-\left(\sin(45)\right)^{2}$$

Add \frac{1666666666666665100573742024936}{1666666666666667189224707921389} and 1.93662912599143627759717608058248 to get \frac{61179773458154915523793644260457672141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}.

$$\frac{61179773458154915523793644260457672141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}-\left(\sin(45)\right)^{2}$$

Get the value of \sin(45) from trigonometric values table.

$$\frac{61179773458154915523793644260457672141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}-\left(\frac{\sqrt{2}}{2}\right)^{2}$$

To raise \frac{\sqrt{2}}{2} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{61179773458154915523793644260457672141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}-\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{61179773458154915523793644260457672141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}-\frac{2}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{61179773458154915523793644260457672141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}-\frac{2}{4}$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{61179773458154915523793644260457672141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}-\frac{1}{2}$$

Subtract \frac{1}{2} from \frac{61179773458154915523793644260457672141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000} to get \frac{50763106791488245591139219751776422141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}.

$$\frac{50763106791488245591139219751776422141814355609657988064633309}{20833333333333339865308849017362500000000000000000000000000000}$$