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4) Express as a single trigonometric function:

 

sin x(tan x + cot x)

 Feb 2, 2020
edited by Guest  Feb 2, 2020
 #1
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=  s  ( s/c   + c/s)

= s^2 / c + c =

   (s^2 + c^2) /c   =   1/c = sec x

 Feb 2, 2020
 #2
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The answerer Guest is correct but let me make it clearer.

 

\(sin(x)(tan(x)+cot(x))\) "Express as a single trigonometric function"

 \(tan(x)=\frac{sin(x)}{cos(x)}\) (1)

\(cot(x)=\frac{cos(x)}{sin(x)}\) (2)

Substitute (1) and (2) into the equation

\(sin(x)(\frac{sin(x)}{cos(x)}+\frac{cos(x)}{sin(x)})\)

Multiply

\((\frac{sin^2(x)}{cos(x)}+cos(x))\). Now write it as a single fraction

\((\frac{sin^2(x)+cos^2(x)}{cos(x)})\)

Pythagorean identity, \(sin^2(x)+cos^2(x)=1\) substitute

\(\frac{1}{cos(x)}\) 

Notice that \(sec(x)=\frac{1}{cos(x)}\)

So \(sin(x)(tan(x)+cot(x))\)\(= sec(x)\)

.
 Feb 3, 2020

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