Question

(x sin a + y cos a)^2 + (x cos a – y sin a)^2 = x^2 + y^2

Answer 1

Proved

Explanation:

Required

Prove that:

`(x\ sin\ a + y\ cos\ a)^2 + (x\ cos\ a - y\ sin\ a)^2 = x^2 + y^2`

Solving from left to right:

Open brackets

`(x\ sin\ a + y\ cos\ a)(x\ sin\ a + y\ cos\ a) + (x\ cos\ a - y\ sin\ a)(x\ cos\ a - y\ sin\ a) = x^2 + y^2`
`x^2\ sin^2 a + 2xy\ sin\ a\ cos\ a + y^2\ cos^2 a + x^2\ cos^2 a - 2xy\ sin\ a\ cos\ a + y^2\ sin^2 a = x^2 + y^2`

Collect Like Terms

`x^2\ sin^2 a + 2xy\ sin\ a\ cos\ a - 2xy\ sin\ a\ cos\ a + y^2\ cos^2 a + x^2\ cos^2 a+ y^2\ sin^2 a = x^2 + y^2`

`x^2\ sin^2 a + y^2\ cos^2 a + x^2\ cos^2 a+ y^2\ sin^2 a = x^2 + y^2`

Collect Like Terms

`x^2\ sin^2 a + y^2\ sin^2 a + x^2\ cos^2 a + y^2\ cos^2 a = x^2 + y^2`

Factorize:

`(x^2 + y^2)\ sin^2 a + (x^2 + y^2) cos^2 a = x^2 + y^2`

Further factorize

`(x^2 + y^2)(sin^2 a + cos^2 a) = x^2 + y^2`

In trigonometry:

`sin^2 a + cos^2 a = 1`

So, we have:

`(x^2 + y^2)(1) = x^2 + y^2`

`x^2 + y^2 = x^2 + y^2`

Proved