1Thanks
• 1 Post By Soroban

Thread:

1. simplifying trig expression (I think)

Not sure if this is a Calc or a trig question. But I wanted to take the derivative of this function:

$y = -csc\,x - sin\,x$

Through simple steps I got:

$-\frac{d}{dx}(csc\,x)\,-\,\frac{d}{dx}sin\,x$

$csc\,x\,cot\,x\,-\,cos\,x$

However the answer in the book is...

$cos\,x\,cot^2\,x$

So presumably my calculus is wrong, or I didn't simplify enough. I have tried various transformations but never quite end up with that. Can anybody show me how to simplify to that?

2. Re: simplifying trig expression (I think)

Hello, infraRed!

$\text{Find the derivative of: }\:y \:=\: -\csc x - \sin x$

$\text{I got: }\:y' \:=\:\csc x\cot x - \cos x$

$\text{However, the answer in the book is: }\,\cos x \cot^2\!x$

$\text{So presumably my calculus is wrong, or I didn't simplify enough.}$

$\csc x\cot x - \cos x \;=\;\csc x\cdot\tfrac{\cos x}{\sin x} - \cos x$

. . . . . . . . . . . . . . . $=\;\cos x\left(\tfrac{\csc x}{\sin x} - 1\right)$

. . . . . . . . . . . . . . . $=\;\cos x\left(\frac{\csc x}{\frac{1}{\csc x}} - 1\right)$

. . . . . . . . . . . . . . . $=\;\cos x\underbrace{\left(\csc^2\!x - 1\right)}_{\text{This is }\cot^2x}$

. . . . . . . . . . . . . . . $=\;\cos x\cot^2\!x$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 13, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8445244431495667, "perplexity_flag": "head"}