1. Find value of `tan(x)+cot(x)`

Solution:
`tan(x)+cot(x)`

`=tan(x)+cot(x)`

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

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

`=(1)/(cos(x)sin(x))`

`=sec(x)csc(x)`

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2. Find value of `tan^2(x)-sin^2(x)`

Solution:
`tan^2(x)-sin^2(x)`

`=tan^(2)(x)-sin^(2)(x)`

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

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

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

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

`=tan^(2)(x)sin^(2)(x)`

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3. Find value of `csc(x)/cos(x)-cos(x)/sin(x)`

Solution:
`csc(x)/cos(x)-cos(x)/sin(x)`

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

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

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

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

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

`=tan(x)`

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4. Find value of `sec(x)/2(tan(x)+cot(x))`

Solution:
`sec(x)/2(tan(x)+cot(x))`

`=(sec(x)(tan(x)+cot(x)))/2`

`=((1/(cos(x)))(((sin(x))/(cos(x)))+((cos(x))/(sin(x)))))/2`

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

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

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

`=(sec^2(x)csc(x))/2`

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5. Find value of `csc(x)/cos(x)-cos(x)/sin(x)`

Solution:
`csc(x)/cos(x)-cos(x)/sin(x)`

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

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

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

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

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

`=tan(x)`

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