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Makovka662 [10]
1 year ago
5

Students were asked to prove the identity (sec x)(csc x) = cot x + tan x. ​

Mathematics
1 answer:
pogonyaev1 year ago
8 0

Let's prove that (sec x)(csc x) is equal to cot x + tan x

\Longrightarrow  \sf (sec(x) )(csc(x))

\Longrightarrow  \sf \dfrac{1}{\cos \left(x\right)\sin \left(x\right)}

\Longrightarrow  \sf \dfrac{\cos ^2\left(x\right)+\sin ^2\left(x\right)}{\cos \left(x\right)\sin \left(x\right)}

\Longrightarrow  \sf \dfrac{\cos ^2\left(x\right)}{\cos \left(x\right)\sin \left(x\right)} + \dfrac{\sin ^2\left(x\right)}{\cos \left(x\right)\sin \left(x\right)}

\Longrightarrow  \sf \dfrac{\cos\left(x\right)}{\sin \left(x\right)} + \dfrac{\sin \left(x\right)}{\cos \left(x\right)}

\Longrightarrow  \sf cot(x)  + tan(x)

Hence student A did correctly prove the identity properly.

Also Looking at student B's work, he verified the identity properly.

So, Both are correct in their own way.

<h3>Part B</h3>

Identities used:

\rightarrow \sf sin^2 (x) + cos^2 (x) = 1       (appeared in step 3)

\sf \rightarrow \dfrac{cos(x) }{sin(x) }  = cot(x)               (appeared in step 6)

\rightarrow \sf \dfrac{sin(x )}{cos(x) }  = tan(x)               (appeared in step 6)

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The question is incomplete. Here is the complete question.

Find the measurements (the lenght L and the width W) of an inscribed rectangle under the line y = -\frac{3}{4}x + 3 with the 1st quadrant of the x & y coordinate system such that the area is maximum. Also, find that maximum area. To get full credit, you must draw the picture of the problem and label the length and the width in terms of x and y.

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