Buktikan tan (45°+a) = cos a + sin a/cos a - sin a = 1 + sin 2a/ cos 2a

Buktikan tan (45°+a) = cos a + sin a/cos a – sin a = 1 + sin 2a/ cos 2a

Jawab:

Penjelasan dengan langkah-langkah:

$tan (45+alpha )\\=frac{tan 45+tan alpha }{1-tan45.tan alpha }\\=frac{1+frac{sinalpha }{cos alpha } }{1-(1)(frac{sinalpha }{cos alpha }) }\\=frac{frac{cos alpha +sin alpha }{cos alpha } }{frac{cos alpha -sinalpha }{cos alpha } } \\=frac{cos alpha +sinalpha }{cosalpha -sinalpha }$

Terbukti $tan (45+alpha )=frac{cos alpha +sinalpha }{cosalpha -sinalpha }$

$frac{cos alpha +sin alpha }{cos alpha -sin alpha } *frac{cos alpha +sinalpha }{cos alpha +sinalpha } \\=frac{cos^2alpha +sin^2alpha +2cosalpha .sinalpha }{cos^2alpha -sin^2alpha } \\=frac{1+sin 2alpha }{cos 2alpha }$

Terbukti $frac{cos alpha +sin alpha }{cosalpha -sinalpha }=frac{1+sin 2alpha }{cos 2alpha }$

$tan (45+alpha )=frac{cos alpha +sinalpha }{cosalpha -sinalpha }$ dan $frac{cos alpha +sin alpha }{cosalpha -sinalpha }=frac{1+sin 2alpha }{cos 2alpha }$ , maka $tan (45+alpha )=frac{1+sin 2alpha }{cos 2alpha }$

Jadi, terbukti $tan (45+alpha )=frac{cos alpha +sinalpha }{cosalpha -sinalpha }=frac{1+sin2alpha }{cos2alpha }$