Question

#QualityQuestion
@Trigonometry


$\bf{Prove ~the~ identity~:}$
(1 + cos(x) + cos(2x)) / (sin(x) + sin(2x)) = cot(x)​

Answer 1

Answer:

1+cos(2x)+cos(x)/sin(x)+2sin(x)+cos(x)=cos(x)/sin(x) 2cossquare X+ cosx/ sinx( 1+2 cosx)=cosx/sinx. cosx(2cosx+1)/ sinx (1+2cosx)= cosx/sinx. cosx/ sinx = cosx/ sinx. = o

Answer 2

Simplifying numerator:

= 1 + cosx + cos2x

= 1 + cosx + 2cos²x - 1

= cosx(1 + 2cosx)

Simplifying denominator:

=> sinx + sin2x

=> sinx + 2sinxcosx

=> sinx(1 + 2cosx)

Therefore,

=> (1 + cosx + cos2x)/(sinx + sin2x)

=> cosx(1 + 2cosx) / sinx(1 + 2cosx)

=> cosx/sinx

=> cotx

Using,

cos2x = 2cos²x - 1 ; sin2x = 2sinxcosx