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Prove that cos(570)sin(510)+sin(330)cos(390)=0

Answer
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Hint: We break the angle values inside the bracket in such a way so we can relate to the quadrants in the plane and write the values for the question in a simpler way. Try to break the values as adding or subtracting from 360,180and relate to the quadrant diagram.
* We know the values of all trigonometric angles are positive in the first quadrant.
Values of only sinθ are positive in the second quadrant.
Values of only tanθ are positive in the third quadrant.
Values of onlycosθ are positive in the fourth quadrant.

Complete step-by-step answer:
We solve for all the trigonometric terms separately.
For better understanding we draw the quadrant division

     
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Firstly we solve for cos(570)
We can write
cos(570)=cos(360+210)
Since 360+ goes to the first quadrant where all trigonometric angles are positive
cos(360+210)=cos(210)
Now we can write
cos(210)=cos(180+30)
Since 180+ goes to the third quadrant where all tan angles are positive, so all cos angles are negative.
 cos(180+30)=cos(30)=32
So, cos(570)=32 … (1)
Now we solve for sin(510)
We can write
sin(510)=sin(360+150)
Since 360+ goes to the first quadrant where all trigonometric angles are positive
sin(360+150)=sin(150)
Now we can write
sin(150)=sin(18030)
Since 180 goes to the second quadrant where all sin angles are positive.
 sin(18030)=sin(30)=12
So, sin(510)=12 … (2)
Now we solve for sin(330)
Since, sin is an odd function therefore, sin(θ)=sinθ
sin(330)=sin(330)
We can write
sin(330)=sin(36030)
Since 360 goes to the fourth quadrant where all cos angles are only positive.
sin(36030)=sin(30)
Since, sin is an odd function therefore, sin(θ)=sinθ
sin(36030)=sin(30)=12
So, sin(330)=12 … (3)
Now we solve for cos(390)
Since, cos is an even function therefore, cos(θ)=cosθ
cos(390)=cos(390)
We can write
cos(390)=cos(360+30)
Since 360+ goes to the first quadrant where all trigonometric angles are positive
cos(360+30)=cos(30)=32
So, cos(390)=32 … (4)
Substitute values from equation (1), (2), (3) and (4) in the following equation
cos(570)sin(510)+sin(330)cos(390)=0
32×12+12×32=034+34=00=0
Therefore, LHS=RHS
Hence Proved

Note: Students can many times make mistakes when the angle between the brackets is negative, so always check first if the trigonometric function alone is an even or odd function. An odd function brings out the negative sign while an even function eradicates the negative sign.