Question

Prave following identities
(x cosA+y sinA)²+(x sinA-y cosA=x²+y²
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Answer 1

To prove the given identity (x cosA+y sinA)²+(x sinA-y cosA)=x²+y², we expand the left side of the equation and simplify it step-by-step to show that it is equal to x² + y².

Step-by-step explanation:

The given expression is (x cosA+y sinA)²+(x sinA-y cosA)=x²+y².

To prove this identity, we expand the left side of the equation:

(x cosA+y sinA)²+(x sinA-y cosA)²

= (x² cos² A + 2xy sinA cosA + y² sin² A) + (x² sin² A - 2xy sinA cosA + y² cos² A)

Simplifying further, we get:

= x² cos² A + x² sin² A + y² sin² A + y² cos² A

= x² (cos² A + sin² A) + y² (sin² A + cos² A)

= x² + y²

This proves the given identity.