1. 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))`

`=0.6124`

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

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

`=0.6124`

Result is proved...

---

2. Prove result `2sin(30)+2tan(45)-3cos(60)-2cos^2(30)=0`

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

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

`=1+2-1.5-1.5`

`=0`

`=`RHS

Result is proved...

---

3. Prove result `sin^2(50)+sin^2(40)=1`

Solution:
LHS `=sin^2(50)+sin^2(40)`

`sin(50)=cos(40)`

`sin(50)=sin(90-40)=cos(40)`

`=cos^2(40)+sin^2(40)`

`=1`

`=`RHS

`:.` Result is proved...

---

4. Prove result `sin(35)*sec(55)-cos(55)*csc(35)=tan(25)-cot(65)`

Solution:
LHS `=sin(35)sec(55)-cos(55)csc(35)`

`sec(55)=csc(35)` and `cos(55)=sin(35)`

`sec(55)=sec(90-35)=csc(35)`

`cos(55)=cos(90-35)=sin(35)`

`=sin(35)csc(35)-sin(35)csc(35)`

`=1-1`

`=0-> (1)`

RHS `=tan(25)-cot(65)`

`cot(65)=tan(25)`

`cot(65)=cot(90-25)=tan(25)`

`=tan(25)-tan(25)`

`=0-> (2)`

from (1) and (2)
Result is proved...

---

5. Prove result `tan(45)*sec(72)*sin(58)=cos(32)*csc(18)`

Solution:
LHS `=tan(45)sec(72)sin(58)`

`sec(72)=csc(18)` and `sin(58)=cos(32)`

`sec(72)=sec(90-18)=csc(18)`

`sin(58)=sin(90-32)=cos(32)`

`=1csc(18)cos(32)`

`=csc(18)cos(32)`

`=(1/(sin(18)))cos(32)`

`=csc(18)cos(32)`

`=`RHS

`:.` Result is proved...

---

6. Prove result `sin^2(34)-cot^2(46)-cos^2(56)+tan^2(44)=sec^2(62)-csc^2(28)`

Solution:
LHS `=sin^2(34)-cot^2(46)-cos^2(56)+tan^2(44)`

`cot(46)=tan(44)` and `cos(56)=sin(34)`

`cot(46)=cot(90-44)=tan(44)`

`cos(56)=cos(90-34)=sin(34)`

`=sin^2(34)-tan^2(44)-sin^2(34)+tan^2(44)`

`=0-> (1)`

RHS `=sec^2(62)-csc^2(28)`

`sec(62)=csc(28)`

`sec(62)=sec(90-28)=csc(28)`

`=csc^2(28)-csc^2(28)`

`=0-> (2)`

from (1) and (2)
Result is proved...

---

---