Question

Please help finals killing me

Answer 1

Solution given:

1.

L.h.s.

cscx+cotx

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

`(1+cosx)/(sinx)`

multiplying and dividing by (1-cosx)

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

`(1-cos²x)/(sinx(1-cosx))`

we have

1-cos²x=sin²x

`(sin²x)/(sinx(1-cosx))`

`(sinx)/(1-cosx)`

=R.h.s.

proved.

2.

l.h.s

`(tan y)/(1-cot y)+(cot y)/(1-tany)`

`((sin y)/(cos y))/(1-(cos x)/(siny))+((cos y)/(sin y))/(1-(siny)/(cos y))`

`((sin y)/(cos y))/((siny -cos y)/(siny))+((cos y)/(sin y))/((cos y-siny)/(cos y))`

`(sin²y)/(cos y(siny-cosy))+(cos ² y)/(sin y(cos y-siny))`

`(sin²y)/(cos y(siny-cosy))- (cos ² y)/(sin y(-cos y+siny))`

`(sin³y -cos³y)/(sinycosy(siny -cosy))`

`((sin y- cos y)(sin²y+sinycosy+cos²y))/(sinycosy(siny -cosy))`

`(sin²y+cos²y +siny cosy)/(sinxcosy)`

`(1+sinxcosy)/(sinycosy)`

`(1)/(sinycosy)+1`

1+cscysecy

R.h.s.

proved.