If `cot^2x=cot(x-y)(x-z),t h e ncot2x` is equal to `(w h e r ex!=pi/4)` . `1/2(tany+tanz)` (b) `1/2(coty+cotz)` `1/2(siny+sinz)` (d) none of these

Options

A. `(1)/(2)(tan y+tanx)`
B. `(1)/(2)(cot y+cot z)`
C. `(1)/(2)(sin y+sinz)`
D. none of these.

Answer

B

Explanation

`cot^(2)x=cot(x-y)cot(x-z)`
or `cot^(2)x=((cot x cot y+1)/(cot y-cotx))((cot x cot z+1)/(cot z-cotx))`
or `cot^(2)xcotycotz-cot^(3)x cot y-cot^(3)x cot z+cot^(4)x`
`=cot^(2)xcotycotz+cotx cot y+cot x cot z+1`
or `cot^(3)x(coty+coty+cotz)+1-cot^(4)x=0`
or `(cot(x(coty+cotz)(1+cot^(2)x)+(1-cot^(2)x)(1+cot^(2)x)=0`
or `[cotx(cot y+cotz)+(1-cot^(2)x)]=0`
or `(cot^(2)x-1)/(2cotx)=(1)/(2)(cot y+cotz)`