How to you simplify cotx+tanx?

Question

How to you simplify cotx+tanx?

2 Answers

Impact of this question

73857 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-12-07T15:40:31

The answer is $= 2 csc (2 x)$ $=2csc(2x)$

Explanation:

We need

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

$cot x = \frac{cos x}{sin x}$ $cotx=cosx/sinx$

${cos}^{2} x + {sin}^{2} s = 1$ $cos^2x+sin^2s=1$

$sin 2 x = 2 sin x cos x$ $sin2x=2sinxcosx$

$csc x = \frac{1}{sin x}$ $cscx=1/sinx$

Therefore,

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

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

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

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

$= 2 csc (2 x)$ $=2csc(2x)$

Answer 2 · 2017-12-07T15:59:09

Alternatively, we can get $sec (x) csc (x)$ $sec(x)csc(x)$

Explanation:

If we write $cot (x)$ $cot(x)$ as $\frac{1}{tan (x)}$ $1/tan(x)$ , we get:
$cot (x) + tan (x) = \frac{1}{tan (x)} + tan (x)$ $cot(x)+tan(x)=1/tan(x)+tan(x)$

Then we bring under a common denominator:
$= \frac{1}{tan (x)} + \frac{tan (x) \cdot tan (x)}{tan (x)} = \frac{1 + {tan}^{2} (x)}{tan (x)}$ $=1/tan(x)+(tan(x)*tan(x))/tan(x)=(1+tan^2(x))/tan(x)$

Now we can use the ${tan}^{2} (x) + 1 = {sec}^{2} (x)$ $tan^2(x)+1=sec^2(x)$ identity:
$= \frac{{sec}^{2} (x)}{tan (x)}$ $=sec^2(x)/tan(x)$

To try and work out some of the relationships between these functions, let's represent the functions in terms of a right triangle. We'll let $a$ $a$ be the adjacent, $b$ $b$ be the opposite and $c$ $c$ be the hypotenuse. This gives:
$= \frac{{(\frac{c}{a})}^{2}}{\frac{b}{a}} = \frac{c^{2}}{a^{2}} \cdot \frac{a}{b} = \frac{c^{2}}{a b} = \frac{c}{a} \cdot \frac{c}{b}$ $=((c/a)^2)/(b/a)=c^2/a^2*a/b=c^2/(ab)=c/a*c/b$

Now if we convert back to trigonometric functions, we get:
$\frac{c}{a} \cdot \frac{c}{b} = sec (x) csc (x)$ $c/a*c/b=sec(x)csc(x)$

This is equal to Narad T's answer of $2 csc (2 x)$ $2csc(2x)$

Science

Math

Humanities

... and beyond

How to you simplify cotx+tanx?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question