Question

Which of the following is not an identity for tan(x/2)

A.
`(1 - cos x)/(sin x)`

B.
`(sin x)/(1 + cos x)`

C.
`(cos x)/(1-sin x)`

D.
`+- sqrt((1 - cosx)/(1 + cos x))`

Answer 1

C and D.

Explanation:

Check each one using x = 60 degrees:

A. tan(x/2) = tan 30 = 0.5774

1 - cos 60 / sin 60 = 0.5774 So A is an identity

B. sin 60 / (1 + cos 60) = 0.5774 :- B is an identity.

C. cos 60 / ( 1 - sin 60) = 3.732 so This is NOT an identity.

D +/-sqrt( (1 - cos60)/(1 + cos 60)) = +/- 0.5774. Because oif the +/- I don't think this is an identity. Sorry I can't be sure.

Answer 2

Options C and D.

Explanation:

A.
`((1-cox))/(sinx)=(1-(2cos^(2)(x)/(2)-1))/(2sin(x)/(2)cos(x)/(2))`

=
`((2-2cos^(2)(x)/(2)))/(2sin(x)/(2)cos(x)/(2))`

=
`(2(1-cos^(2)(x)/(2)))/(2sin(x)/(2)cos(x)/(2))`

=
`(2sin^(2)(x)/(2))/(2sin(x)/(2)cos(x)/(2))`

=
`(sin(x)/(2) )/(cos(x)/(2))`

=
`tan(x)/(2)`

Therefore, it's an identity for
`tan(x)/(2)`

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

`=(sin(x)/(2))/(cos(x)/(2))`

`=tan(x)/(2)`

Therefore, it's an identity
`tan(x)/(2)`

C.
`(cosx)/(1-sinx)=(cos^(2)(x)/(2)-sin^(2)(x)/(2))/(2sin(x)/(2)cos(x)/(2))`

`=(cos(x)/(2))/(2sin(x)/(2))-(sin(x)/(2))/(2cos(x)/(2))`

`=(1)/(2)[cot(x)/(2)-tan(x)/(2)]`

Therefore, it's not an identity for
`tan(x)/(2)`

D.
`\pm \sqrt{(1-cosx)/(1+cosx)}=\pm {\sqrt{(1-(1-2sin^(2)(x)/(2)))/(1+(2cos^(2)(x)/(2)-1))}}`

`=\pm \sqrt{(sin^(2)(x)/(2))/(cos^(2)(x)/(2))}`

`=\pm \sqrt{tan^(2)(x)/(2)}`

`=\pm tan(x)/(2)`

Therefore, it's not an identity for
`tan(x)/(2)`

Options C and D are not the identities.