Q51 of 268 Page 5

If a cos θ – b sinθ = x and a sinθ + b cosθ = y that a2 + b2 = x2 + y2.

Taking RHS =x2 + y2

Putting the values of x and y, we get


(a cos θ – b sin θ)2 + (a sin θ + b cos θ)2


= a2cos2θ + b2 sin2θ – 2ab cos θ sin θ + a2sin2θ + b2 cos2θ + 2ab cos θ sin θ


= a2 (cos2 θ + sin2 θ) + b2 (cos2 θ + sin2 θ)


= a2 + b2 [∵ cos2 θ + sin2 θ = 1]


=RHS


Hence Proved


More from this chapter

All 268 →
49

If a cos θ + b sinθ = c, then prove that a sinθ – b cos θ = ±

50

If 1+sin2θ = 3 sinθ . cosθ, then prove that tan θ = 1 or 1/2.

 52

If x = a sec θ + b tan θ a and y = a tan θ + b sec θ, then prove that x2 – y2 = a2 – b2.

 53

If (a2 – b2) sin θ + 2ab cosθ = a2 + b2, then prove that .