How can this be reduced to the simplest form?

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How can this be reduced to the simplest form?

$\frac{1 + {tan}^{2} x}{1 + {cot}^{2} x}$ $(1+tan^2x) / (1+cot^2x)$

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Answer 1 · 2018-03-06T18:14:42

See explanation

Explanation:

We want to simplify

$\frac{1 + {tan}^{2} (x)}{1 + {cot}^{2} (x)}$ $(1+tan^2(x))/(1+cot^2(x))$

Use the pythagorean trig identity

${sin}^{2} (x) + {cos}^{2} (x) = 1$ $sin^2(x)+cos^2(x)=1$

$\Rightarrow 1 + {cot}^{2} (x) = {csc}^{2} (x) \leftarrow divided by {sin}^{2} (x)$ $=>1+cot^2(x)=csc^2(x)color(green)(larr "divided by" sin^2(x))$

$\Rightarrow 1 + {tan}^{2} (x) = {sec}^{2} (x) \leftarrow divided by {cos}^{2} (x)$ $=>1+tan^2(x)=sec^2(x)color(green)(larr "divided by" cos^2(x))$

Thus

$\frac{1 + {tan}^{2} (x)}{1 + {cot}^{2} (x)} = \frac{{sec}^{2} (x)}{{csc}^{2} (x)} = \frac{{sin}^{2} (x)}{{cos}^{2} (x)} = {tan}^{2} (x)$ $(1+tan^2(x))/(1+cot^2(x))=sec^2(x)/csc^2(x)=sin^2(x)/cos^2(x)=tan^2(x)$

Answer 2 · 2018-03-06T18:19:52

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

Explanation:

$using the trigonometric identities$ $"using the "color(blue)"trigonometric identities"$

$∙ x 1 + {tan}^{2} x = {sec}^{2} x$ $•color(white)(x)1+tan^2x=sec^2x$

$∙ x 1 + {cot}^{2} x = {csc}^{2} x$ $•color(white)(x)1+cot^2x=csc^2x$

$∙ x sec x = \frac{1}{cos x} and csc x = \frac{1}{sin x}$ $•color(white)(x)secx=1/cosx" and "cscx=1/sinx$

$\Rightarrow \frac{1 + {tan}^{2} x}{1 + {cot}^{2} x}$ $rArr(1+tan^2x)/(1+cot^2x)$

$= \frac{{sec}^{2} x}{{csc}^{2} x}$ $=sec^2x/csc^2x$

$= \frac{\frac{1}{{cos}^{2} x}}{\frac{1}{{sin}^{2} x}}$ $=(1/cos^2x)/(1/sin^2x)$

$= \frac{1}{{cos}^{2} x} \times {sin}^{2} x$ $=1/cos^2x xxsin^2x$

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

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

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How can this be reduced to the simplest form?

$\frac{1 + {tan}^{2} x}{1 + {cot}^{2} x}$ $(1+tan^2x) / (1+cot^2x)$

2 Answers

Explanation:

Explanation:

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$\frac{1 + {tan}^{2} x}{1 + {cot}^{2} x}$ $(1+tan^2x) / (1+cot^2x)$