Question

Prove? (sin4a+2cos3a-sin2a)/(cos4a-2sin3a-cos2a)=-ctg3a

Answer 1

Please see below.

Explanation:

We know that,
`color(red)((1)sinC-sinD=2cos((C+D)/2)sin((C-D)/2)`
`color(blue)((2)cosC-cosD=-2sin((C+D)/2)sin((C-D)/2)`
`color(violet)((3)costheta/sintheta=cottheta=ctg theta`

Here,

`(sin4a+2cos3a-sin2a)/(cos4a-2sin3a-cos2a)=-ctg3a`

We take,

`LHS=(sin4a+2cos3a-sin2a)/(cos4a-2sin3a-cos2a)`

`color(white)(LHS)=(color(red)(sin4a-sin2a)+2cos3a)/(color(blue)(cos4a-cos2a)-2sin3a)...toApply(1)and(2)`

#color(white)(LHS)=(color(red)(2cos((4a+2a)/2)sin((4a- 2a)/2))+2cos3a)/(color(blue)(-2sin((4a+2a)/2)sin((4a-2a)/2))-2sin3a)#

`color(white)(LHS)=-[(2cos3asina+2cos3a)/(2sin3asina+2sin3a)]`

#color(white)(LHS)=- [(2cos3acancel((sina+1)))/(2sin3acancel((sina+1)))]#

`color(white)(LHS)=-(cos3a)/(sin3a)...tocolor(violet)(Apply(3)`

`color(white)(LHS)=-cot3a`

`LHS=RHS`