Question

if cosΦ - sinΦ = √2sinΦ , prove that cosΦ + sinΦ = √2sinΦ

Answer 1

$cos - sin = \sqrt{2} sin$

Now square on both side

$(cos - sin) {}^{2} = ( \sqrt{2} sin) {}^{2}$

$cos {}^{2} + sin {}^{2} - 2sincos = 2sin {}^{2}$

$- 2sincos = 2sin {}^{2} - sin {}^{2} - cos {}^{2}$

$- 2sincos = sin {}^{2} - cos {}^{2}$

Take - common from RHS


$- 2sincos = - (cos {}^{2} - sin {}^{2} )$

$2sincos = (cos + sin)(cos - sin)$

$now \: put \: cos - sin = \sqrt{2} sin$

$2sincos \div \sqrt{2} sin = cos + sin$

$cos + sin = \sqrt{2} cos$


Thanks