Mathematics
If x cos A + y sin A = m and x sin A - y cos A = n, then prove that :
x2 + y2 = m2 + n2
Answer
To prove:
x2 + y2 = m2 + n2
Substituting value of m and n in R.H.S. of the equation :
= (x cos A + y sin A)2 + (x sin A - y cos A)2
= x2 cos2 A + y2 sin2 A + 2xy cos A sin A + x2 sin2 A + y2 cos2 A - 2xy sin A cos A
= x2 cos2 A + x2 sin2 A + y2 cos2 A + y2 sin2 A
= x2(sin2 A + cos2 A) + y2(sin2 A + cos2 A)
By formula,
sin2 A + cos2 A = 1
⇒ x2 × 1 + y2 × 1
⇒ x2 + y2.
Since, L.H.S. = R.H.S.
Hence, proved that x2 + y2 = m2 + n2.