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Prove the following identity. Please be clear and neat. SHOW ALL STEPS and number your steps if necessary. Work with one side to manipulate it (use identities, simplify, etc.) t…

Prove the following identity. Please be clear and neat. SHOW ALL STEPS and number your steps if necessary. Work with one side to manipulate it (use identities, simplify, etc.) t…
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4 votes
Prove the following identity. Please be clear and neat. SHOW ALL STEPS and number your steps if necessary. Work with one side to manipulate it (use identities, simplify, etc.) to get it to match the other side (do not move parts over the = sign).

Prove the following identity. Please be clear and neat. SHOW ALL STEPS and number-example-1
asked
User Dmitry Negoda
by
9.2k points

1 Answer

5 votes

Answer:

Step-by-step explanation:

We are given the identity:


(\sec x)/(cotx +tanx)=\sin x

We would show a prove of this identity as shown below:


\begin{gathered} (\sec x)/(cotx +tanx)=\sin x \\ \text{Let's define the identities, we have:} \\ \sec x=(1)/(\cos x),\cot x=(\cos x)/(\sin x),\tan x=(\sin x)/(\cos x) \\ \Rightarrow((1)/(\cos x))/((\cos x)/(\sin x)+(\sin x)/(\cos x)) \\ \text{Multiply the numerator \& denominator by the common factor of the denominatro, we have:} \\ ((1)/(\cos x))/((\cos x)/(\sin x)+(\sin x)/(\cos x))*(\cos x\cdot\sin x)/(\cos x\cdot\sin x) \\ (\sin x)/((\cos x)^2+(\sin x)^2) \\ (\sin x)/(\cos ^2x+\sin ^2x) \\ But\colon\cos ^2x+\sin ^2x=1 \\ \Rightarrow(\sin x)/(1)=\sin x \\ =\sin x \\ \\ \therefore(\sec x)/(cotx+tanx)=\sin x(PROVEN) \end{gathered}

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User Piotr Niewinski
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