Tolong buktikan identitas trigonometri berikut;

tan^{2}x +sin^{2}x = (sec x +cos x) (sec x – cos x)

Tolong buktikan identitas trigonometri berikut;

Jawaban Terkonfirmasi

$tan^{2}x$ + $sin^{2} x$ = (sec x+cos x) (sec x – cos x)
olah pada ruas kiri
$frac{ sin^{2}x }{ cos^{2} x}$ + $frac{ sin^{2}x }{1}$
samakan penyebut
$frac{ sin^{2}x + sin^{2}x . cos^{2}x }{ cos^{2}x } = frac{ sin^{2}x (1+ cos^{2}x) }{ cos^{2}x }$
= $frac{ (1- cos^{2} x) (1+ cos^{2}x )}{ cos^{2}x } = frac{1- cos^{4}x }{ cos^{2}x }$
= $frac{1}{ cos^{2}x} - frac{ cos^{4}x }{ cos^{2}x } = sec^{2}x - cos^{2}x$
= (sec x + cos x) (sec x – cos x)

Tan^2(x) + sin^2(x) = $frac{sin ^{2}x }{cos ^{2}x } + frac{sin ^{2}x }{cos ^{2}x } cos ^{2}x$
= $frac{sin ^{2}x(1+cos ^{2} x) }{cos ^{2} x}$
= $frac{(1-cos ^{2} x)(1+cos ^{2} x)}{cos ^{2} x}$
= $frac{1-cos ^{4} x}{cos ^{2} x}$
= $frac{1}{cos ^{2} x} -cos ^{2} x$
= $sec ^{2} x-cos ^{2} x$
=(sec x + cos x)(sec x – cos x)