Respuesta :

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sin x and csc x are reciprocals, so sin x ( csc x) = 1.

Next:  

sin x    -cos x
------ * ----------- * cos x = - (cos x)^2
   1        sin x

Then we have 1 - (cos x)^2, which in turn is equal to (sin x)^2 (answer)

Answer:

The expression sinx(cscx-cotx cosx) can be simplified to sin²x.

Step-by-step explanation:

Consider the provided trigonometric expression.

[tex]sinx(cscx-cotx cosx)[/tex]

Open the parentheses and apply the distributive property: a(b+c)=ab+ac

[tex]sinx\cdot cscx-cotx \cdot cosx \cdot sinx[/tex]

Now use the identity: [tex]cscx= \frac{1}{sinx}, cotx=\frac{cosx}{sinx}[/tex]

[tex]sinx\cdot \frac{1}{sinx}-\frac{cosx}{sinx} \cdot cosx \cdot sinx [/tex]

[tex]1-cos^2x[/tex]

Use the identity: 1 - cos²x = sin²x

Thus, [tex]1-cos^2x=sin^2x[/tex]

Hence the expression sinx(cscx-cotx cosx) can be simplified to sin²x.

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