I'm trying to prove the following problem: \(\displaystyle{\frac{{{{\sec}^{{2}}{\left({x}\right)}}}}{{{\cot{{\left({x}\right)}}}}}}-{{\tan}^{{3}}{\left({x}\right)}}={\tan{{\left({x}\right)}}}\)

Bryson Whitney

Bryson Whitney

Answered question

2022-04-07

I'm trying to prove the following problem:
sec2(x)cot(x)tan3(x)=tan(x)

Answer & Explanation

Mey9ci0

Mey9ci0

Beginner2022-04-08Added 14 answers

We will work on the left side and show it is equl to tan(x). Let's change everything into sin(x) and cos(x) using the following formulas
sec2(x)=1cos2(x), cot(x)=cos(x)sin(x)
tan3(x)=sin3(x)cos3(x)
This will give us:
1cos2(x)cos(x)sin(x)sin3(x)cos3(x)
Cleaning up the left fraction will give you
sin(x)cos3(x)sin3(x)cos3(x)
From here, factor out sin(x) from the top to get:
sin(x)(1sin2(x))cos3(x)
Finally, use the Pythagorean Identity cos2(x)=1sin2(x) to finish it off.

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