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)`
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)`
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)`
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)`
This material is intended as a summary. Use your textbook for detail explanation.
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