# what does the expression equal to? `[cos 10+sin 40]/[sin 10+cos 40]`

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Apparently we don't want to use decimal approximation from a calculator, we need to start by using some trig identities to simplify.

`[cos10+sin40]/[sin10+cos40]=`

`[cos(25-15)+sin(25+15)]/[sin(25-150+cos(25+15)]=`

`[cos25cos15+sin25sin15+sin25cos15+cos25sin15]/[sin25cos15-cos25sin15+cos25cos15-sin25sin15]=`

`[cos25(cos15+sin15)+sin25(sin15+cos15)]/[sin25(cos15-sin15)+cos25(cos15-sin15)]=`

`[(sin15+cos15)(cos25+sin25)]/[(cos15-sin15)(sin25+cos25)]`

`[sin15+cos15]/[cos15-sin15]`

``I am going to multiply both numerator and denominator by

cos15-sin15, we get

`[cos^2(15)-sin^2(15)]/[(cos15-sin15)^2]`

`=[cos30]/[cos^2(15)+sin^2(15)-2cos15sin15]=`

`[cos30]/[1-sin30]=`

`[(sqrt3)/2]/[1-(1/2)]=sqrt3/(2-1)=sqrt3`

Sorry, I mean cos 10 + sin 40 / sin 10 + cos 40