Question

rewrite each expression such that the argument x is positive. (10 points) a) cot(x)cos(−x) sin(x) b) cos(x) tan(x)sin(−x)

Answer 1

a) To rewrite the expression
`\displaystyle\sf \cot(x)\cos(-x)\sin(x)` such that the argument
`\displaystyle\sf x` is positive, we can use the following trigonometric identities:

`\displaystyle\sf \cot(x) = (1)/(\tan(x)) \quad \text{and} \quad \cos(-x) = \cos(x)`.

Applying these identities, we can rewrite the expression as:

`\displaystyle\sf (\cos(x))/(\tan(x)) \cdot \cos(x) \cdot \sin(x)`.

Simplifying further:

`\displaystyle\sf \cos(x) \cdot \cos(x) \cdot \sin(x) = \cos^(2)(x) \sin(x)`.

Therefore, the expression
`\displaystyle\sf \cot(x)\cos(-x)\sin(x)`, rewritten such that
`\displaystyle\sf x` is positive, is
`\displaystyle\sf \cos^(2)(x) \sin(x)`.

b) To rewrite the expression
`\displaystyle\sf \cos(x) \tan(x) \sin(-x)` such that
`\displaystyle\sf x` is positive, we can use the following trigonometric identity:

`\displaystyle\sf \sin(-x) = -\sin(x)`.

Applying this identity, we can rewrite the expression as:

`\displaystyle\sf \cos(x) \tan(x) \cdot (-\sin(x))`.

Simplifying further:

`\displaystyle\sf -\cos(x) \sin(x) \tan(x)`.

Therefore, the expression
`\displaystyle\sf \cos(x) \tan(x) \sin(-x)`, rewritten such that
`\displaystyle\sf x` is positive, is
`\displaystyle\sf -\cos(x) \sin(x) \tan(x)`.

`\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}`

♥️
`\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}`