Question

How i prove this ?

Answer 1

See below

Explanation:

Take into account that `15º=45-30`

Appliying `sin(a-b)=sinacosb-cosasinb` and `cos(a-b)=cosacosb+senasenb`

We have `sin15=sin(45-30)=sin45cos30-cos45 sen30=sqrt2/2·sqrt3/2-sqrt2/2·1/2=sqrt6/4-sqrt2/4`

By the other hand..`cos15=cos(45-30)=cos45cos30+sin45sin30=sqrt2/2·sqrt3/2+sqrt2/2·1/2=sqrt6/4+sqrt2/4`

Finally we have `sin15+tan30cos15=sqrt6/4-sqrt2/2+sqrt3/3(sqrt6/4+sqrt2/4)=sqrt6/4-sqrt2/4+sqrt18/12+sqrt6/12=3sqrt6/12-3sqrt2/12+sqrt18/12+sqrt6/12=1/12(3sqrt6-cancel(3sqrt2)+cancel(3sqrt2)+sqrt6)=4sqrt6/12=sqrt6/3`

Answer 2

As proved

Explanation:

`sin 15 + tan 30 cos 15`

`sin 15+ (sin 30 / cos 30 ) * cos 15, " as " tan theta = (sin theta / cos theta)`

`=> (cos 30 sin 15 + sin 30 cos 15) / cos 30, " taking cos 30 as L C M"`

`sin A cos B + cos A sin B = sin (A + B)`

`:. =>sin (30 + 15) / cos 30`

`=> sin 45 / cos 30`

We know `sin 45 = 1/ sqrt2, cos 30 = sqrt 3 / 2`

`"Hence " => (1/sqrt2) / (sqrt3 / 2) =2 / (sqrt 2 * sqrt 3)`

`=> cancel(2)^color(red)(sqrt2) / (cancelsqrt 2 * sqrt3 ) * (sqrt 3 / sqrt3), " multiply and divide by " sqrt3`

`=>(sqrt 2 * sqrt 3) / 3 = sqrt6 / 3`

Q E D