1. Find value of `sin(30)*cos(60)+sin(30)*cos(60)`

Solution:
`sin(30)*cos(60)+sin(30)*cos(60)`

`=sin(30)cos(60)+sin(30)cos(60)`

`=(1/2)*(1/2)+(1/2)*(1/2)`

`=1/4+1/4`

`=1/2`

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2. Find value of `sin(30)*csc(30)-sin(60)*csc(60)`

Solution:
`sin(30)*csc(30)-sin(60)*csc(60)`

`=sin(30)csc(30)-sin(60)csc(60)`

`=(1/2)*(2)-(sqrt(3)/2)*(2/sqrt(3))`

`=1-1`

`=0`

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3. Find value of `(2tan(60))/(1+tan^2(60))`

Solution:
`(2tan(60))/(1+tan^2(60))`

`=(2tan(60))/(1+tan^2(60))`

`2tan(60)=2sqrt(3)`

`=2tan(60)`

`=2*(sqrt(3))`

`=2sqrt(3)`

`1+tan^2(60)=4`

`=1+tan^2(60)`

`=1+(sqrt(3)^2)`

`=1+3`

`=4`

`=(2sqrt(3))/(4)`

`=(sqrt(3))/2`

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4. Find value of `(3sec(60)+2tan(45)+csc(30))/(sin^2(60)+cot^2(45))`

Solution:
`(3sec(60)+2tan(45)+csc(30))/(sin^2(60)+cot^2(45))`

`=(3sec(60)+2tan(45)+csc(30))/(sin^2(60)+cot^2(45))`

`3sec(60)+2tan(45)+csc(30)=10`

`=3sec(60)+2tan(45)+csc(30)`

`=3*(2)+2*(1)+(2)`

`=6+2+2`

`=10`

`sin^2(60)+cot^2(45)=7/4`

`=sin^2(60)+cot^2(45)`

`=((sqrt(3)/2)^2)+(1^2)`

`=3/4+1`

`=7/4`

`=(10)/(7/4)`

`=40/7`

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5. Find value of `2sin(30)+2tan(45)-3cos(60)-2cos^2(30)`

Solution:
`2sin(30)+2tan(45)-3cos(60)-2cos^2(30)`

`=2sin(30)+2tan(45)-3cos(60)-2cos^2(30)`

`=2*(1/2)+2*(1)-3*(1/2)-2*((sqrt(3)/2)^2)`

`=1+2+(-3)/2+(-3)/2`

`=0`

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6. Prove result `sin(30)*cos(45)*tan(60)=sin(45)*cos(60)*cot(30)`

Solution:
LHS `=sin(30)cos(45)tan(60)`

`=(1/2)*(1/sqrt(2))*(sqrt(3))`

`=(sqrt(3))/(2sqrt(2))`

RHS `=sin(45)cos(60)cot(30)`

`=(1/sqrt(2))*(1/2)*(sqrt(3))`

`=(sqrt(3))/(2sqrt(2))`

Result is proved...

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