taylor csc^2(x)

Topic

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`x`²	`x`^□	log_□	√☐	^□√☐	≤	≥	□□	·	÷	`x`^◦	π
(☐)^′	`ddx`	∂∂`x`	∫	∫_□^□	lim	∑	∞	`θ`	(`f` ◦ `g`)	`f`(`x`)

☐²

x^☐

√☐

^□√☐

□□

log_□

θ

∞

∫

ddx

≥	≤	·	÷	`x`^◦	(☐)	\|☐\|	(`f` ◦ `g`)	`f`(`x`)	ln	`e`^☐
(☐)^′	∂∂`x`	∫_□^□	lim	∑	sin	cos	tan	cot	csc	sec

simplify solve for inverse tangent line

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taylor csc²(x)

Solution

csc²(1)−2csc²(1)cot(1)(x−1)+(2csc²(1)cot²(1)+csc⁴(1))(x−1)²−4cot³(1)csc²(1)+8csc⁴(1)cot(1)3 (x−1)³−−44csc⁴(1)cot²(1)−8cot⁴(1)csc²(1)−8csc⁶(1)12 (x−1)⁴+…

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Solution steps

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Taylor Series

Taylor series of function f(x) at a is defined as:
f(x)=f(a)+f^′(a)1! (x−a)+f^′′(a)2! (x−a)²+f^′′′(a)3! (x−a)³+…

Apply the Taylor Formula

=csc²(1)+ddx (csc²(x))(1)1! (x−1)+d²dx² (csc²(x))(1)2! (x−1)²+d³dx³ (csc²(x))(1)3! (x−1)³+…

Evaluate Derivatives

=csc²(1)+−2csc²(1)cot(1)1! (x−1)+−2(−2csc²(1)cot²(1)−csc⁴(1))2! (x−1)²+−2(4cot³(1)csc²(1)+8csc⁴(1)cot(1))3! (x−1)³+−2(−44csc⁴(1)cot²(1)−8cot⁴(1)csc²(1)−8csc⁶(1))4! (x−1)⁴+…

Refine

=csc²(1)−2csc²(1)cot(1)(x−1)+(2csc²(1)cot²(1)+csc⁴(1))(x−1)²−4cot³(1)csc²(1)+8csc⁴(1)cot(1)3 (x−1)³−−44csc⁴(1)cot²(1)−8cot⁴(1)csc²(1)−8csc⁶(1)12 (x−1)⁴+…