simplify (sec^2x)\(tan x +cot^2 x tan x)?

Answer 1

I assume that by "\" you meant "/".

Simplify

(sec² x)/{tan x + (cot² x)(tan x)}
= (sec² x)/{tan x + cot x}
= (1/cos² x)/{(sin x)/(cos x) + (cos x)/(sin x)}

Multiply numerator and denominator thru by (cos² x)(sin x).

= sin x/{(sin² x)(cos x) + (cos³ x)}
= sin x/{(cos x)[(sin² x) + (cos² x)]}
= sin x/cos x
= tan x

Answer 2

Let's first simplify the second paranthesis. You have:

tan x + ((cot x)^2) * tan x =

= tan x (1 + (cos x)^2/(sin x)^2 ) =

= (tan x) * (1/(sin x)^2) =

= (sin x / cos x) * (1 / (sin x)^2) =

= 1 / (cos x * sin x)

Now divide the result obtained for the second paranthesis with the expression in the first paranthesis and you should get:

(sec x)^2 * (cos x * sin x) =

= (1 / (cos x)^2)* (sin x * cos x) =

= tan x

Answer 3

(sec(x)^2)/(tan(x) + tan(x)cot(x)^2)
(sec(x)^2)/(tan(x) + tan(x)(1/tan(x))^2)
(sec(x)^2)/(tan(x) + (tan(x)/tan(x)^2))
(sec(x)^2)/(tan(x) + (1/tan(x)))
(sec(x)^2)/((tan(x)^2 + 1)/(tan(x)))
((sec(x)^2)/1)*(tan(x)/(tan(x)^2 + 1)))
(tan(x)sec(x)^2)/(tan(x)^2 + 1)
(tan(x)(tan(x)^2 + 1)))/(tan(x)^2 + 1)
tan(x)

(sec(x)^2)/(tan(x) + tan(x)cot(x)^2) = tan(x)

Answer 4

sec^2x/(tanx+cot^2xtanx)=sec^2x/tanx(1+cot^2x)
we factorise by tanx:
we know that:
cot^2x=1/tan^2x and sec^2x=1+tan^2x
sec^2x/(tanx/tan^2x)/(tan^2x+1)]= sec^2x/tanx*sec^2x
simplify by sec^2x
then,the result is 1/tanx or cotx
(sec^2x)/(tanx+cot^2x tanx) =1/tanx or cotx