Verify the identity: (Picture) Thanks for your time

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vkinard | High School Teacher | (Level 2) Adjunct Educator

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There are often multiple ways to verify trig identities.  Here is an alternative way to approach the problem.  Here is another option:

`(cos^3x*sec^2x+tanx*sinx)/(cotx*sinx)*cscx` 

First substitute for a couple of the reciprocal and quotient identities:

`(cos^3x*(1/(cos^2x))+(sinx/cosx)*sinx)/(cosx/sinx*sinx)*cscx`

Then we can cancel and combine the smaller fractions:

`(cosx+(sin^2x)/cosx)/cosx*cscx`

We can then divide each term in the numerator by the denominator:

`(cosx/cosx+((sin^2x)/cosx)/cosx)*cscx`

We can then simplify each of these fractions:

`(1+(sin^2x)/(cos^2x))*cscx`

We can then use one of the quotient identities again:

`(1+tan^2x)*cscx`

And finally we will use one of the Pythagorean identities:

`sec^2x*cscx`

`cscx*sec^2x`

 

 

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