Question

Cos(x) /sin(pi/2 +x) + sin(-x) /sin (pi+x) - tan (pi/2+2) / cot(x) =3

Answer 1

sin(pi/2+X)=cosX
sin(-X)=-sinX
sin(pi+X)=-sinX
tan(pi/2+X)=-cotX
putting the values we will get thr above result

Answer 2

The solution to the proof of cos (x) / sin (pi/2 + x) + sin (-x) / sin (pi+x) - tan (pi/2+2) / cot (x)  = 3 is given below.

Given: cos (x) / sin (pi/2 + x) + sin (-x) / sin (pi+x) - tan (pi/2+2) / cot (x)  = 3

To Find: Prove the above expression.

Solution:

To solve this numerical, we need to first know a few prerequisites such as;

• In the first quadrant, the values for sin, cos, and tan are positive.
• In the second quadrant, the values for sin are positive only while the rest are negative.
• In the third quadrant, the values for tan are positive only while the rest are negative.
• In the fourth quadrant, the values for cos are positive only while the rest are negative.

Coming to the numerical, the question given is;

            cos (x) / sin (pi/2 + x) + sin (-x) / sin (pi+x) - tan (pi/2+2) / cot (x)

Let us evaluate each of these individually,

cos (x): It is in the first quadrant, so its value is positive and shall remain as cos x

sin ( π/2 + x ): It is in the second quadrant, so it will become, cos x

sin (-x): the sin of any negative value is negative, so it becomes - sin x

sin ( π + x ): It is in the third quadrant, so it becomes - sin x

tan ( π/2 + x ): It is in the second quadrant, so it will become, - cot x

cot x: It is in the first quadrant, so its value is positive and shall remain as cot x

Employing all the values in the original question, we get;

  cos (x) / sin (pi/2 + x) + sin (-x) / sin (pi+x) - tan (pi/2+2) / cot (x)

⇒ ( cos x / cos x ) + ( sin x / sin x ) - ( - cot x / cot x )

⇒ 1 + 1 + 1

⇒ 3

Hence, it is proved.

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