Question

if cosФ+sinФ=√2cosФ , then prove that cosФ-sinФ=√2sinФ

Answer 1

(cosФ+sinФ)²=(√2cosФ)²cos²Ф+sin²Ф+2sinФcosФ=2cos²Ф
2cos²Ф -cos²Ф-sin²Ф=2sinФcosФ
cos²Ф-sin²Ф=2sinФcosФ
(cosФ+sinФ)(cosФ-sinФ)=2sinФcosФ
(cosФ-sinФ)=2sinФcosФ/√2cosФ       (since cosФ+ sinФ=√2cosФ)
cosФ-sinФ=√2sinФ

Answer 2

sinФ+cosФ=√2cosФ
sinФ=√2cosФ-cosФ
=cosФ(√2-1)
multipying by (√2+1)on both sides
sinФ(√2+1)=cosФ(√2-1)(√2+1)
√2sinФ+sinФ=cosФ(2-1)
cosФ-sinФ=√2sinФ