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# 4) Express as a single trigonometric function: Show all Work

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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 $$sin(x)(tan(x)+cot(x))$$$$= sec(x)$$

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Feb 3, 2020