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Question
Evaluate the following:
`(cos 45°)/(sec 30° + cosec 30°)`
Solution
`(cos 45°)/(sec 30° + cosec 30°)`
= `(1/sqrt2)/(2/sqrt3+2)`
= `(1/sqrt2)/((2 + 2sqrt3)/sqrt3)`
= `sqrt3/(sqrt2(2+2sqrt3))`
= `sqrt3/(2sqrt2+2sqrt6)`
= `(sqrt3(2sqrt6-2sqrt2))/(((2sqrt6)+2sqrt2)(2sqrt6-2sqrt2))`
= `(2sqrt3(sqrt6-sqrt2))/((2sqrt6)^2 - (2sqrt2)^2)`
= `(2sqrt3(sqrt6-sqrt2))/(24-8)`
= `(2sqrt3(sqrt6-sqrt2))/16`
= `(sqrt18-sqrt6)/8`
= `(3sqrt2 - sqrt6)/8`
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