If x cos A + y sin A = m and x sin A - y cos A = n, then prove that : x 2 + y 2 = m 2 + n 2

To prove: x 2 + y 2 = m 2 + n 2 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 = x 2 cos 2 A + y 2 sin 2 A + 2xy cos A sin A + x 2 sin 2 A + y 2 cos 2 A - 2xy sin A cos A = x 2 cos 2 A + x 2 sin 2 A + y 2 cos 2 A + y 2 sin 2 A = x 2 (sin 2 A + cos 2 A) + y 2 (sin 2 A + cos 2 A) By formula, sin 2 A + cos 2 A = 1 ⇒ x 2 × 1 + y 2 × 1 ⇒ x 2 + y 2 . Since, L.H.S. = R.H.S. Hence, proved that x 2 + y 2 = m 2 + n 2 .

Mathematics

If x cos A + y sin A = m and x sin A - y cos A = n, then prove that :
x² + y² = m² + n²

Trigonometric Identities

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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.

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Mathematics

If x cos A + y sin A = m and x sin A - y cos A = n, then prove that :
x² + y² = m² + n²

Trigonometric Identities

Answer

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