Question

Prove the following trig identities

sin 2x + cos 2x / sin x + cos x = 2cosx

2sinxcosx / cos⁴x-sin⁴x = tan2x

Answer 1

(b)

$\frac{2sinxcosx}{cos^4x-sin^4x}=tan2x$

taking LHS

$=\frac{sin2x}{(cos^2x+sin^2x)(cos^2x-sin^2x)}\\=\frac{sin2x}{cos^2x-sin^2x}\\=\frac{sin2x}{cos2x}\\=tan2x$

(a)

$\frac{sin2x+cos2x}{cosx+sinx}=2cosx$

taking LHS

$=\frac{2sinxcosx+2cos^2x-1}{cosx+sinx}$

$2cosx- \frac{1}{sinx+cosx}$

so sinx +cosx cannnot be zero so this is not solved further