Question

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

Answer 1

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 2

Answer:

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

Step-by-step explanation:

Consider the provided trigonometric expression.

$sinx(cscx-cotx cosx)$

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

$sinx\cdot cscx-cotx \cdot cosx \cdot sinx$

Now use the identity: $cscx= \frac{1}{sinx}, cotx=\frac{cosx}{sinx}$

$sinx\cdot \frac{1}{sinx}-\frac{cosx}{sinx} \cdot cosx \cdot sinx$

$1-cos^2x$

Use the identity: 1 - cos²x = sin²x

Thus, $1-cos^2x=sin^2x$

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