Question

If `sin(x+20^0)=2sinxcos40^0,where in (0,pi/2),` then which of the following hold(s) good? (a)`cos2x=1/2` (b) `cos e c4x=2` (c)`secx/2=sqrt(6)-sqrt(2)` (d) `tanx/2=(2-sqrt(3))`

Answer 1

`sinxcos20+cosxsin20=2sinxcos40`
`cosxsin20=sinx(2cos40-cos20)`
`tanx=(sin20)/(cos40+(cos40-cos20))`
`=(sin20)/(cos40+2sin30*sin(-10)`
`=(sin20)/(sin50-sin10)`
`=sin20/(2cos30sin20)`
`tanx=1/sqrt3`
`x=30`
`sec15=1/cos15=(2sqrt2)/(sqrt3+1)*(sqrt3-1)/(sqrt3-1)`
`=sqrt6-sqrt2`
`cos15=cos(45-30)`
`=cos45*cos30+sin45*sin30`
`=1/sqrt2*sqrt3/2+1/sqrt2*1/2`
`cos15=(sqrt3+1)/(2sqrt2)`
`sin15=sin(45-30)=1/sqrt2*sqrt3/2-1/sqrt2*1/2=(sqrt3-1)/(2sqrt2)`
`tann15=(sqrt3-1)/(sqrt3+1)*(sqrt3-1)/(sqrt3-1)`
`=(3+1-2sqrt3)/(3-1)=2sqrt3-1`.