Question

if acos theta - bsin theta = x and asin theta + bcos theta = y, prove that a2 + b2 = x2 + y2

Answer 1

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Answer 2

Answer:

Given,

$a cos \theta - b sin \theta = x$ -------(1)

$a sin \theta + b cos \theta = y$ ------(2),

We have to prove that,

$a^2+b^2=x^2+y^2$

( Equation (1) )² + ( Equation (2) )²,

$(a cos \theta - b sin \theta)^2+(a sin \theta + b cos \theta)^2=x^2+y^2$

$a^2 cos^2\theta + b^2 sin \theta - 2ab sin \theta cos \theta + a^2 sin^2 \theta + b^2 cos^2 \theta + 2 ab sin \theta cos \theta=x^2+y^2$

$a^2(cos^2\theta + sin^2 \theta)+b^2(sin^2 \theta + cos^2\theta)=x^2+b^2$

$a^2+b^2=x^2+y^2$

Hence, proved.