# PROVING TRIGONOMETRIC PROBLEMS WORKSHEET WITH ANSWERS

Proving Trigonometric Problems Worksheet With Answers :

Here we are going to see how to prove trigonometric problems.

## Proving Trigonometric Problems Worksheet With Answers - Questions

Question 1 :

Show that cos2 A + cos2 B − 2 cosAcosB cos(A + B) = sin2(A + B).

Solution :

L.H.S

=  cos2 A + cos2 B − 2 cosAcosB cos(A + B)

= cos2 A + cos2 B − 2 cosAcosB [cos A cos B - sin A sin B]

= cos2 A + cos2 B−2 cos2Acos2B + 2 cosA cosB sin A sin B

=  cos2A - cos2A cos2cos2 B − cos2A cos2B + 2cosA cosB sinA sin B

From 1st and 2nd term, factor out cos2A

From 3rd and 4th term, factor out cos2B

=  cos2A (1 - cos2B) cos2 B(1 - cos2A) + 2cosA cosB sinA sin B

=  cos2A (sin2B) cos2 B(sin2A) + 2cosA cosB sinA sin B

=  (cosA sinB)2 + (cosB sinA)2 + 2cosA cosB sinA sin B

=  (cosA sinB + cosB sinA)2

=  [sin (A + B)]2

=  sin2(A + B)

Hence proved.

Question 2 :

If cos(α − β) + cos(β − γ) + cos(γ − α) = −3/2, then prove that  cos α + cos β + cos γ = sinα + sinβ + sinγ = 0.

Solution :

cos(α − β)  =  cos α cos β - sin α sin β   ----(1)

cos(β − γ)  =  cos β cos γ - sin β sin γ   ----(2)

cos(γ − α)  =  cos γ cos α - sin γ sin α    ----(3)

(1) + (2) + (3)

cos(α − β) + cos(β − γ) + cos(γ − α)

=  cos α cos β + sin α sin β + cos β cos γ + sin β sin γ + cos γ cos α + sin γ sin α

So, cos α cos β + sin α sin β + cos β cos γ + sin β sin γ + cos γ cos α + sin γ sin α   =  -3/2

2[cos α cos β + sin α sin β + cos β cos γ + sin β sin γ + cos γ cos α + sin γ sin α] + 3  =  0

2cos α cos β + 2sin α sin β + 2cos β cos γ + 2sin β sin γ + 2cos γ cos α + 2sin γ sin α] + 3  =  0

2cos α cos β + 2sin α sin β + 2cos β cos γ + 2sin β sin γ + 2cos γ cos α + 2sin γ sin α] + (cos2α + sin2α) + (cos2β + sin2β) + (cos2γ + sin2γ)  =  0

cos2α + cos2β + cos2γ + 2cos α cos β + 2cos β cos γ + 2cos γ cos α + sin2α + sin2β + sin2γ + 2sin α sin β + 2sin β sin γ + 2sin γ sin α  =  0

(cosα + cosβ + cosγ)2 + (sinα + sinβ + sinγ) =  0

cosα + cosβ + cosγ  =  0

sinα + sinβ + sinγ  =  0

Hence proved. After having gone through the stuff given above, we hope that the students would have understood how to do proving trigonometric problems.

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