How to prove the following identities cos(30°)-cos(60°)\cos(30°)+sin(60°)=tan(15°)?

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Answer 1 · 2018-08-12T18:29:07

A Proof is given in The Explanation.

#color(red){"The Correct Problem is to prove that,"}#

#color(red)[(cos30^@-cos60^@)/(cos30^@+cos60^@)=tan15^@]#.

#"The Expression"=(cos30^@-cos60^@)/(cos30^@+cos60^@)#,

#={-2sin((30^@+60^@)/2)sin((30^@-60^@)/2)}/{2cos((30^@+60^@)/2)cos((30^@-60^@)/2)}#,

#={-2sin45^@sin(-15^@)}/{2cos45^@cos(-15^@)}#,

#=(1/sqrt2*sin15^@)/(1/sqrt2*cos15^@)#,

#=tan15^@#, as desired!