Simplifying trignometric equations, proving identities and evaluating functions sin(30)*cos(60)+sin(30)*cos(60) like Example

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

Solution:

sin(30)⋅cos(60)+sin(30)⋅cos(60) $sin(30)*cos(60)+sin(30)*cos(60)$

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

=(12)⋅(12)+(12)⋅(12) $=(1/2)*(1/2)+(1/2)*(1/2)$

=14+14 $=1/4+1/4$

=12 $=1/2$

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

Solution:

sin(30)⋅csc(30)-sin(60)⋅csc(60) $sin(30)*csc(30)-sin(60)*csc(60)$

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

=(12)⋅(2)-(√32)⋅(2√3) $=(1/2)*(2)-(sqrt(3)/2)*(2/sqrt(3))$

=1-1 $=1-1$

=0 $=0$

3. Find value of 2tan(60)1+tan2(60) $(2tan(60))/(1+tan^2(60))$

Solution:

2tan(60)1+tan2(60) $(2tan(60))/(1+tan^2(60))$

=2tan(60)1+tan2(60) $=(2tan(60))/(1+tan^2(60))$

2tan(60)=2√3 $2tan(60)=2sqrt(3)$

=2tan(60) $=2tan(60)$

=2⋅(√3) $=2*(sqrt(3))$

=2√3 $=2sqrt(3)$

1+tan2(60)=4 $1+tan^2(60)=4$

=1+tan2(60) $=1+tan^2(60)$

=1+(√32) $=1+(sqrt(3)^2)$

=1+3 $=1+3$

=4 $=4$

=2√34 $=(2sqrt(3))/(4)$

=√32 $=(sqrt(3))/2$

4. Find value of 3sec(60)+2tan(45)+csc(30)sin2(60)+cot2(45) $(3sec(60)+2tan(45)+csc(30))/(sin^2(60)+cot^2(45))$

Solution:

3sec(60)+2tan(45)+csc(30)sin2(60)+cot2(45) $(3sec(60)+2tan(45)+csc(30))/(sin^2(60)+cot^2(45))$

=3sec(60)+2tan(45)+csc(30)sin2(60)+cot2(45) $=(3sec(60)+2tan(45)+csc(30))/(sin^2(60)+cot^2(45))$

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

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

=3⋅(2)+2⋅(1)+(2) $=3*(2)+2*(1)+(2)$

=6+2+2 $=6+2+2$

=10 $=10$

sin2(60)+cot2(45)=74 $sin^2(60)+cot^2(45)=7/4$

=sin2(60)+cot2(45) $=sin^2(60)+cot^2(45)$

=((√32)2)+(12) $=((sqrt(3)/2)^2)+(1^2)$

=34+1 $=3/4+1$

=74 $=7/4$

=1074 $=(10)/(7/4)$

=407 $=40/7$

5. Find value of 2sin(30)+2tan(45)-3cos(60)-2cos2(30) $2sin(30)+2tan(45)-3cos(60)-2cos^2(30)$

Solution:

2sin(30)+2tan(45)-3cos(60)-2cos2(30) $2sin(30)+2tan(45)-3cos(60)-2cos^2(30)$

=2sin(30)+2tan(45)-3cos(60)-2cos2(30) $=2sin(30)+2tan(45)-3cos(60)-2cos^2(30)$

=2⋅(12)+2⋅(1)-3⋅(12)-2⋅((√32)2) $=2*(1/2)+2*(1)-3*(1/2)-2*((sqrt(3)/2)^2)$

=1+2+-32+-32 $=1+2+(-3)/2+(-3)/2$

=0 $=0$

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

Solution:
LHS

=sin(30)cos(45)tan(60) $=sin(30)cos(45)tan(60)$

=(12)⋅(1√2)⋅(√3) $=(1/2)*(1/sqrt(2))*(sqrt(3))$

=√32√2 $=(sqrt(3))/(2sqrt(2))$

RHS

=sin(45)cos(60)cot(30) $=sin(45)cos(60)cot(30)$

=(1√2)⋅(12)⋅(√3) $=(1/sqrt(2))*(1/2)*(sqrt(3))$

=√32√2 $=(sqrt(3))/(2sqrt(2))$

Result is proved...

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1. sin(30)⋅cos(60)+sin(30)⋅cos(60)sin(30)*cos(60)+sin(30)*cos(60) like Example

1. sin(30)⋅cos(60)+sin(30)⋅cos(60) $sin(30)cos(60)+sin(30)cos(60)$ like Example