Question

Can anyone help me with these? I don’t think I did them correctly

Answer 1

1.
`sec(x)-1`

2.
`2 sec^2(x)`

3.
`sec(x)-1`

Explanation:

1.
`tan(x)[csc(x) - cot(x)]`

expand the brackets:

`\implies tan(x)csc(x)-tan(x)cot(x)`

Substitute the following equivalents:

`tan(x)=(sin(x))/(cos(x))`,
`csc(x)=(1)/(sin(x))`,
`cot(x)=(1)/(tan(x))`

`\implies (sin(x))/(cos(x)) \cdot (1)/(sin(x))-(tan(x))/(1) \cdot (1)/(tan(x))`

Cancel the common factors:

`\implies (1)/(cos(x))-1`

`\implies sec(x)-1`

2.
`(1)/(sin(x)+1)-(1)/(sin(x)-1)`

`\implies ((sin(x)-1)-(sin(x)+1))/((sin(x)+1)(sin(x)-1))`

`\implies (-2)/(sin^2(x)-1)`

Using identity
`sin^2(x)+cos^2(x)=1`

`\implies (-2)/(-cos^2(x))`

`\implies (2)/(cos^2(x))`

`\implies 2 sec^2(x)`

3.
`(tan^2(x))/(sec(x)+1)`

Using identity
`1 + tan^2(x)=sec^2(x)`:

`\implies (sec^2(x)-1)/(sec(x)+1)`

`\implies ((sec(x)+1)(sec(x)-1))/(sec(x)+1)`

Cancel common factor
`sec(x)+1`:

`\implies sec(x)-1`