Simplify each of the following using the trig identities. $({sin}^{2} x) + ({sin}^{2} x) ({tan}^{2} x)$ $(sin^2x)+(sin^2x)(tan^2x)$ ?

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Answer 1 · 2018-04-11T01:32:16

${sin}^{2} x + {sin}^{2} x {tan}^{2} x = {tan}^{2} x$ $sin^2x+sin^2xtan^2x=tan^2x$

Simplify: ${sin}^{2} x + {sin}^{2} x {tan}^{2} x$ $sin^2x+sin^2xtan^2x$

First, factor out ${sin}^{2} x$ $sin^2x$ from the expression:

${sin}^{2} x (1 + {tan}^{2} x)$ $sin^2x(1+tan^2x)$

Now we can use this trig identity

$1 + {tan}^{2} x = {sec}^{2} x$ $1+tan^2x=sec^2x$

Now we have

${sin}^{2} x {sec}^{2} x$ $sin^2xsec^2x$

We know that

$sec x = \frac{1}{cos x}$ $secx=1/cosx$

So it is then true that

${sec}^{2} x = \frac{1}{{cos}^{2} x}$ $sec^2x=1/cos^2x$

Now we have

$\frac{{sin}^{2} x}{{cos}^{2} x}$ $sin^2x/cos^2x$

We know that

$tan x = \frac{sin x}{cos x}$ $tanx=sinx/cosx$

So it is then true that

${tan}^{2} x = \frac{{sin}^{2} x}{{cos}^{2} x}$ $tan^2x=sin^2x/cos^2x$

So for the final answer we have is

${tan}^{2} x$ $tan^2x$

Simplify each of the following using the trig identities. (sin2x)+(sin2x)(tan2x)(sin^2x)+(sin^2x)(tan^2x)?