Question

What is the step-by-step of 1/cos(x) csc^2(x) = sin(x) / cot(x)?

a) 1
b) csc^2(x)
c) cos(x) / sin(x)
d) cot(x) / sin(x)

Answer 1

The step-by-step simplification of
`\( (1)/(\cos(x) \csc^2(x)) = (\sin(x))/(\cot(x)) \)` results in
`\( (\sin^2(x))/(\cos(x)) \)` , which is equivalent to option c) cos(x) / sin(x). Thus the correct option C.

Step-by-step explanation:

To simplify the expression
`\( (1)/(\cos(x) \csc^2(x)) = (\sin(x))/(\cot(x)) \)`, we can start by expressing the cosecant
`(\(\csc(x)\)) in terms of sine (\(\sin(x)\))` and cosine
`(\(\cos(x)\))`.

First, rewrite the denominator using the reciprocal identity
`\( \csc(x) = (1)/(\sin(x)) \)`:

`\[ (1)/(\cos(x) \left((1)/(\sin(x))\right)^2) \]`

Next, simplify the expression by multiplying through by the reciprocal:

`\[ (\sin^2(x))/(\cos(x)) \]`

Now, the expression on the right side is
`\( (\sin(x))/(\cot(x)) \)` . Recall that
`\( \cot(x) = (\cos(x))/(\sin(x)) \)`. Substituting this into the expression, we get:

`\[ (\sin^2(x))/(\cos(x)) = (\sin(x))/(\cot(x)) \]`

So, the simplified form is
`\( (\sin(x))/(\cot(x)) \)` , which is equivalent to option c) cos(x) / sin(x).