Question

Verify the identity.

cotangent of x to the second power divided by quantity cosecant of x plus one equals quantity one minus sine of x divided by sine of x

Answer 1

Therefore,

`(\cot^(2)x)/(\csc x +1)=(1-\sin x)/(\sin x)` ....Proved

Explanation:

To Prove:

`(\cot^(2)x)/(\csc x +1)=(1-\sin x)/(\sin x)`

Proof:

Left Hand Side =
`(\cot^(2)x)/(\csc x +1)`

Using Identity
`\cot^(2)x=\csc^(2)-1`

=
`(\csc^(2)x-1)/(\csc x +1)`

Using identity a² - b² =( a - b )( a + b )

=
`((\csc x-1)(\csc x+1))/(\csc x +1)`

=
`(\csc x-1)`

Now Using identity
`\csc x=(1)/(\sin x)` we get

=
`((1)/(\sin x)-1)`

=
`((1-\sin x)/(\sin x))`

=Right Hand Side

Therefore,

`(\cot^(2)x)/(\csc x +1)=(1-\sin x)/(\sin x)` ....Proved