Question

If the side of a right angled triangle are {cos2α + cos2β + 2cos(α+β)} and {sin2α + sin2β + 2sin(α+β)}, then the length of the hypotenuse is

(A) 2[1 + cos(α - β)]

(B) 2[1 + cos(α + β)]

(C) 4cos²(α - β)/2

(D) 4sin²(α + β)/2

Answer 1

Correct option (A, C)

Explanation:

Hypotenuse

= √([cos2α + cos2β + 2cos(α+β)]² + [sin2α + sin2β + 2sin(α+β)]²)

cos2α + cos2β + 2cos(α + β) 

= cos²α – sin²α + cos²β – sin²b + 2cosacosβ – 2sinasinb 

= (cosα + cosβ)² – (sinα + sinβ)²sin2α + sin2β + 2sin(α + β) 

= 2sinacosα + 2sinbcosβ + 2sinacosb + 2cosasinb 

= 2sina[cosα + cosb] + 2sinb[cosβ + cosa] 

= 2(cosα + cosβ)(sinα + sinβ) 

Hypotenuse 

= (cosα + cosβ)² + (sinα + sinβ)² 

= 2 + 2cos² acosβ + 2sinbcosb

= 2 + 2cos(α - β) = 4cos²(α - β)/2