How do you verify $tan(x)sin(x) + cos(x)= sec(x)$ ?

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Answer 1 · 2016-04-14T01:22:43

Recall the following quotient, Pythagorean, and reciprocal identities:

$1. color(red)(tanx=sinx/cosx)$

$2. color(darkorange)(sin^2x+cos^2x=1)$

$3. color(blue)(secx=1/cosx)$

$1$ . To verify the given identity, start by working on the left side. Rewrite $tanx$ in terms of $sinx$ and $cosx$ .

Left side:

$color(red)tanxsinx+cosx$

$=color(red)(sinx/cosx)*sinx+cosx$

$2$ . Simplify.

$=sin^2x/cosx+cosx$

$=color(darkorange)(sin^2x+cos^2x)/cosx$

$3$ . Simplify the numerator.

$=color(darkorange)1/cosx$

$=color(green)(|bar(ul(color(white)(a/a)secxcolor(white)(a/a)|)))$

How do you verify tan(x)sin(x) + cos(x)= sec(x) tan(x)sin(x) + cos(x)= sec(x) ?