1. Find value of `cot^4(x)+cot^2(x)`

Solution:
`cot^4(x)+cot^2(x)`

`=cot^4(x)+cot^2(x)`

`=cot^2(x)(cot^2(x)+1)`

`=cot^2(x)(csc^2(x))`

`=cot^2(x)csc^2(x)`

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2. Find value of `1+1/tan^2(x)`

Solution:
`1+1/tan^2(x)`

`=1+1/(tan^2(x))`

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

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

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

`=csc^2(x)`

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3. Find value of `(1+cos(x))/sin(x)+sin(x)/(1+cos(x))`

Solution:
`(1+cos(x))/sin(x)+sin(x)/(1+cos(x))`

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

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

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

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

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

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

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

`=2/(sin(x))`

`=2csc(x)`

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4. Find value of `sin^2(x)cos(x)+sin^3(x)+cos^2(x)sin(x)+cos^3(x)`

Solution:
`sin^2(x)cos(x)+sin^3(x)+cos^2(x)sin(x)+cos^3(x)`

`=sin^2(x)cos(x)+sin^3(x)+cos^2(x)sin(x)+cos^3(x)`

`=(sin^3(x)+cos^3(x))+(sin^2(x)cos(x)+cos^2(x)sin(x))`

`=(sin^3(x)+cos^3(x))+sin(x)cos(x)(sin(x)+cos(x))`

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

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

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

`=(sin(x)+cos(x))*1`

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

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

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