Solve (sin47/cos43)^2+(cos43/sin47)^2-4cos^245

$$( \frac { \sin 47 } { \cos 43 } ) ^ { 2 } + ( \frac { \cos 43 } { \sin 47 } ) ^ { 2 } - 4 \cos ^ { 2 } 45$$

Evaluate

$$\frac{213951294748825099187513307994921}{2860947282793664653902041934291031420896760104334858049421612900}\approx 7.478337543 \cdot 10^{-32}$$

Solution Steps

Evaluate trigonometric functions in the problem

$${(\frac{0.7313537016191705}{0.7313537016191706})} ^ {2} + {(\frac{0.7313537016191706}{0.7313537016191705})} ^ {2} - 4 \cos^{2}(45)$$

Expand \frac{0.7313537016191705}{0.7313537016191706} by multiplying both numerator and the denominator by 10000000000000000.

$$\left(\frac{7313537016191705}{7313537016191706}\right)^{2}+\left(\frac{0.7313537016191706}{0.7313537016191705}\right)^{2}-4\left(\cos(45)\right)^{2}$$

Calculate \frac{7313537016191705}{7313537016191706} to the power of 2 and get \frac{53487823687206267483341310807025}{53487823687206282110415343190436}.

$$\frac{53487823687206267483341310807025}{53487823687206282110415343190436}+\left(\frac{0.7313537016191706}{0.7313537016191705}\right)^{2}-4\left(\cos(45)\right)^{2}$$

Expand \frac{0.7313537016191706}{0.7313537016191705} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{53487823687206267483341310807025}{53487823687206282110415343190436}+\left(\frac{7313537016191706}{7313537016191705}\right)^{2}-4\left(\cos(45)\right)^{2}$$

Calculate \frac{7313537016191706}{7313537016191705} to the power of 2 and get \frac{53487823687206282110415343190436}{53487823687206267483341310807025}.

$$\frac{53487823687206267483341310807025}{53487823687206282110415343190436}+\frac{53487823687206282110415343190436}{53487823687206267483341310807025}-4\left(\cos(45)\right)^{2}$$

Add \frac{53487823687206267483341310807025}{53487823687206282110415343190436} and \frac{53487823687206282110415343190436}{53487823687206267483341310807025} to get \frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}.

$$\frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}-4\left(\cos(45)\right)^{2}$$

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

$$\frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}-4\times \left(\frac{\sqrt{2}}{2}\right)^{2}$$

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

$$\frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}-4\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}$$

Express 4\times \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} as a single fraction.

$$\frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}-\frac{4\left(\sqrt{2}\right)^{2}}{2^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}-\frac{4\times 2}{2^{2}}$$

Multiply 4 and 2 to get 8.

$$\frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}-\frac{8}{2^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}-\frac{8}{4}$$

Divide 8 by 4 to get 2.

$$\frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900}-2$$

Subtract 2 from \frac{5721894565587329307804083868582276793088269033768903612151220721}{2860947282793664653902041934291031420896760104334858049421612900} to get \frac{213951294748825099187513307994921}{2860947282793664653902041934291031420896760104334858049421612900}.

$$\frac{213951294748825099187513307994921}{2860947282793664653902041934291031420896760104334858049421612900}$$