Question

Solve please..........

Answer 1

we are given

`(tanx)/(secx-1)- (sinx)/(1+cosx)=2cot(x)`

Firstly , we will simplify left side

and make it equal to right side

Left side:

`(tanx)/(secx-1)- (sinx)/(1+cosx)`

we know that

sec(x)=1/cos(x)

`(tanx)/((1/cosx)-1)- (sinx)/(1+cosx)`

`(tanx*cosx)/(1-cos(x))- (sinx)/(1+cosx)`

`(sin(x))/(1-cos(x))- (sinx)/(1+cosx)`

now, we can take common denominator

`(sin(x)(1+cosx))/((1-cos(x))(1+cosx))- (sinx(1-cosx))/((1+cosx)(1-cosx))`

`(sin(x)(1+cosx))/(1-cos^2(x))- (sinx(1-cosx))/(1-cos^2x)`

`(sin(x)(1+cosx))/(sin^2x)- (sinx(1-cosx))/(sin^2x)`

now, we can cancel sinx

`((1+cosx))/(sinx)- ((1-cosx))/(sinx)`

`((1+cosx)-(1-cosx))/(sinx)`

now, we can simplify it

`(1+cosx-1+cosx)/(sinx)`

`(2cosx)/(sinx)`

so, we get

`2cot(x)`

we can see that left side is equal to right side

`(tanx)/(secx-1)- (sinx)/(1+cosx)=2cot(x)`............Answer