Question

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

Answer 1

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}`