Solve 1/left(cos(10)right)^2+1/left(sin(20)right)^2+1/left(sin(40)right)^2-1/left(cos(45)right)^2

$$\frac{ 1 }{ { \left( \cos ( 10 ) \right) }^{ 2 } } + \frac{ 1 }{ { \left( \sin ( 20 ) \right) }^{ 2 } } + \frac{ 1 }{ { \left( \sin ( 40 ) \right) }^{ 2 } } - \frac{ 1 }{ { \left( \cos ( 45 ) \right) }^{ 2 } }$$

Evaluate

$$\frac{18310546874999996369796781576184030509627698762784584447936053315862248746293853208975610622}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}\approx 10$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{1}{0.984807753012208 ^ {2}} + \frac{1}{0.3420201433256687 ^ {2}} + \frac{1}{0.6427876096865393 ^ {2}} - \frac{1}{{(\cos^{2}(45))}}$$

Calculate 0.984807753012208 to the power of 2 and get 0.969846310392954075097397035264.

$$\frac{1}{0.969846310392954075097397035264}+\frac{1}{0.3420201433256687^{2}}+\frac{1}{0.6427876096865393^{2}}-\frac{1}{\left(\cos(45)\right)^{2}}$$

Expand \frac{1}{0.969846310392954075097397035264} by multiplying both numerator and the denominator by 1000000000000000000000000000000.

$$\frac{1000000000000000000000000000000}{969846310392954075097397035264}+\frac{1}{0.3420201433256687^{2}}+\frac{1}{0.6427876096865393^{2}}-\frac{1}{\left(\cos(45)\right)^{2}}$$

Reduce the fraction \frac{1000000000000000000000000000000}{969846310392954075097397035264} to lowest terms by extracting and canceling out 256.

$$\frac{3906250000000000000000000000}{3788462149972476855849207169}+\frac{1}{0.3420201433256687^{2}}+\frac{1}{0.6427876096865393^{2}}-\frac{1}{\left(\cos(45)\right)^{2}}$$

Calculate 0.3420201433256687 to the power of 2 and get 0.11697777844051095979530830215969.

$$\frac{3906250000000000000000000000}{3788462149972476855849207169}+\frac{1}{0.11697777844051095979530830215969}+\frac{1}{0.6427876096865393^{2}}-\frac{1}{\left(\cos(45)\right)^{2}}$$

Expand \frac{1}{0.11697777844051095979530830215969} by multiplying both numerator and the denominator by 100000000000000000000000000000000.

$$\frac{3906250000000000000000000000}{3788462149972476855849207169}+\frac{100000000000000000000000000000000}{11697777844051095979530830215969}+\frac{1}{0.6427876096865393^{2}}-\frac{1}{\left(\cos(45)\right)^{2}}$$

Add \frac{3906250000000000000000000000}{3788462149972476855849207169} and \frac{100000000000000000000000000000000}{11697777844051095979530830215969} to get \frac{424540659700572279254963022431128906250000000000000000000000}{44316588600974220158623956135758892709470162173417693081761}.

$$\frac{424540659700572279254963022431128906250000000000000000000000}{44316588600974220158623956135758892709470162173417693081761}+\frac{1}{0.6427876096865393^{2}}-\frac{1}{\left(\cos(45)\right)^{2}}$$

Calculate 0.6427876096865393 to the power of 2 and get 0.41317591116653479173440361044449.

$$\frac{424540659700572279254963022431128906250000000000000000000000}{44316588600974220158623956135758892709470162173417693081761}+\frac{1}{0.41317591116653479173440361044449}-\frac{1}{\left(\cos(45)\right)^{2}}$$

Expand \frac{1}{0.41317591116653479173440361044449} by multiplying both numerator and the denominator by 100000000000000000000000000000000.

$$\frac{424540659700572279254963022431128906250000000000000000000000}{44316588600974220158623956135758892709470162173417693081761}+\frac{100000000000000000000000000000000}{41317591116653479173440361044449}-\frac{1}{\left(\cos(45)\right)^{2}}$$

Add \frac{424540659700572279254963022431128906250000000000000000000000}{44316588600974220158623956135758892709470162173417693081761} and \frac{100000000000000000000000000000000}{41317591116653479173440361044449} to get \frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}.

$$\frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}-\frac{1}{\left(\cos(45)\right)^{2}}$$

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

$$\frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}-\frac{1}{\left(\frac{\sqrt{2}}{2}\right)^{2}}$$

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

$$\frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}-\frac{1}{\frac{\left(\sqrt{2}\right)^{2}}{2^{2}}}$$

Divide 1 by \frac{\left(\sqrt{2}\right)^{2}}{2^{2}} by multiplying 1 by the reciprocal of \frac{\left(\sqrt{2}\right)^{2}}{2^{2}}.

$$\frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}-\frac{2^{2}}{\left(\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}-\frac{4}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}-\frac{4}{2}$$

Divide 4 by 2 to get 2.

$$\frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}-2$$

Subtract 2 from \frac{21972656249999994920716082457075879474212419671579059579752505003906250000000000000000000000}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689} to get \frac{18310546874999996369796781576184030509627698762784584447936053315862248746293853208975610622}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}.

$$\frac{18310546874999996369796781576184030509627698762784584447936053315862248746293853208975610622}{1831054687499999275459650440445924482292360454397237565908225844022000626853073395512194689}$$