Question

How do you simplify this trigonometric expression? 5Calpha%29-cos%28%5Calpha%29%7D" id="TexFormula1" title="\frac{tan(\alpha) }{sec(\alpha)-cos(\alpha)}" alt="\frac{tan(\alpha) }{sec(\alpha)-cos(\alpha)}" align="absmiddle" class="latex-formula">

Answer 1

Answer:

csc(α)

Step-by-step explanation:

We are given $\frac{tan(\alpha) }{sec(\alpha)-cos(\alpha ) }$.

One key trick when dealing with trig is to write all the functions in terms of cosine and sine.

Tangent (tan) is sine / cosine, secant (sec) is 1 / cosine. So, replace these:

$\frac{tan(\alpha) }{sec(\alpha)-cos(\alpha ) }=\frac{\frac{sin(\alpha )}{cos(\alpha )} }{\frac{1}{cos(\alpha) }-cos(\alpha ) }$

In the denominator, let's find a common denominator and subtract those:

$\frac{1}{cos(\alpha )} -cos(\alpha )=\frac{1}{cos(\alpha) } -\frac{cos^{2} (\alpha) }{cos(\alpha) } =\frac{1-cos^2(\alpha) }{cos(\alpha) }$

Remember the trig identity that sin²(α) + cos²(α) = 1, so we know that 1 - cos²(α) = sin²(α). Plug this into the equation:

$\frac{1-cos^2(\alpha) }{cos(\alpha) }=\frac{sin^2(\alpha )}{cos(\alpha) }$

We now have [sin(α)/cos(α)] / [sin²(α)/cos(α)]. The cos(α) in the top and bottom cancel out, and we are left with sin(α) / sin²(α) = 1 / sin(α).

Remember that cosecant is the opposite of sine, so 1/sin(α) = csc(α).

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