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I need help with this practice problem solving I will send two additional pictures that go with this, one is the rest of the question and the other is the answer options

I need help with this practice problem solving I will send two additional pictures-example-1
User Pmichna
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1 Answer

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Given:


(\sec x\sin x)/(\tan x+\cot x)=\sin^2x

Find-: Prove the trigonometric identity.

Sol:

Use some trigonometric formula:


\begin{gathered} \sec x=(1)/(\cos x) \\ \\ \tan x=(\sin x)/(\cos x) \\ \\ \cot x=(\cos x)/(\sin x) \end{gathered}

So,


\begin{gathered} (\sec x\sin x)/(\tan x+\cot x)=\sin^2x \\ \\ ((1)/(\cos x)\cdot\sin x)/((\sin x)/(\cos x)+(\cos x)/(\sin x)).................\text{ First option } \end{gathered}

Then,


=((\sin x)/(\cos x))/((\sin^2x)/(\sin x\cos x)+(\cos^2x)/(\sin x\cos))................................(\text{ Second option\rparen}


=((\sin x)/(\cos x))/((\sin^2x+\cos^2x)/(\sin x\cos x))...........................(\text{ Third option\rparen}

Solve the identity then,


=((\sin x)/(\cos x))/((1)/(\sin x\cos x))..................(\text{ Fourth option\rparen}

Here, use the formula:


\sin^2x+\cos^2x=1

Then,


\begin{gathered} =(\sin x)/(\cos x)\sin x\cos x..........(\text{ Fifth option\rparen} \\ \\ =\sin^2x \end{gathered}

User MHebes
by
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