Q28 of 46 Page 1

If tan x = n tan y and sin x = m sin y, prove that


OR


If x sin3θ + y cos3θ = sinθ cosθ and x sinθ = y cosθ, prove that x2 + y2 = 1.


Given: tan x = n tan y


…(i)


and sin x = m sin y


…(ii)


To Prove:


Proof: Taking RHS



[given]





[∵ 1 + tan2θ = sec2θ]








[∵cos2θ + sin2θ = 1]




= cos2x


= LHS


∴ LHS = RHS


Hence Proved


OR


Given: x sin3θ + y cos3θ = sinθ cosθ and x sinθ = y cosθ


To Prove: x2 + y2 = 1


Proof: Using given equation


x sin3θ + y cos3θ = sinθ cosθ


⇒ xsinθ (sin2θ) + (y cosθ)cos2θ = sinθcosθ


⇒ xsinθ(sin2θ)+(xsinθ)cos2θ = sinθcosθ


[∵, x sinθ = y cosθ]


Taking common (xsinθ)


⇒ xsinθ(sin2θ + cos2θ) = sinθ cosθ


We know that,


sin2θ + cos2θ = 1


⇒ xsinθ(1) = sinθ cosθ


Case1:


⇒ x sinθ = sinθ cosθ


⇒ x = cos θ


Case2:


⇒ ycosθ = sinθ cosθ [∵, x sinθ = y cosθ]


⇒ y = sinθ


Taking LHS


= x2 + y2


Putting the value of x and y, we get


= (cosθ)2 + (sinθ)2


= cos2θ + sin2θ


= 1


= RHS


∴ LHS = RHS


Hence Proved


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