SOLUTION: How do you simplify the following equation with trigonometry identities? tanθ/1-cos^2θ

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Question 1206148: How do you simplify the following equation with trigonometry identities?
tanθ/1-cos^2θ
Found 2 solutions by MathLover1, math_tutor2020:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
tan%28theta%29%2F%281-cos%5E2%28theta%29%29
use identity: sin%5E2%28theta%29=1-cos%5E2%28theta%29 and tan%28theta%29=sin%28theta%29%2Fcos%28theta%29

=%28sin%28theta%29%2Fcos%28theta%29%29%2Fsin%5E2%28theta%29...simplify

=1%2F%28cos%28theta%29%2Asin%28theta%29%29

=%281%2Fcos%28theta%29%29%281%2Fsin%28theta%29%29... use identity:1%2Fcos%28theta%29=sec%28theta%29 and 1%2Fsin%28theta%29=csc%28theta%29

=sec%28theta%29%2Acsc%28theta%29

Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

%28tan%28theta%29%29%2F%281+-+cos%5E2%28theta%29%29

= %28tan%28theta%29%29%2F%28sin%5E2%28theta%29%29 Use the pythagorean trig identity

= matrix%281%2C3%2Ctan%28theta%29%2C%22divide%22%2Csin%5E2%28theta%29%29

= matrix%281%2C3%2Csin%28theta%29%2Fcos%28theta%29%2C%22divide%22%2Csin%5E2%28theta%29%29

= %28sin%28theta%29%2Fcos%28theta%29%29%2A%281%2F%28sin%5E2%28theta%29%29%29

= %281%2Fcos%28theta%29%29%2A%281%2Fsin%28theta%29%29

= sec%28theta%29%2Acsc%28theta%29

Therefore,
%28tan%28theta%29%29%2F%281+-+cos%5E2%28theta%29%29+=+sec%28theta%29%2Acsc%28theta%29
is an identity.

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Alternatively,

%281%2Fcos%28theta%29%29%2A%281%2Fsin%28theta%29%29

= 1%2F%28cos%28theta%29%2Asin%28theta%29%29%29

= 1%2F%280.5sin%282theta%29%29%29 Use a variation of sin(2theta) = 2*sin(theta)*cos(theta)

= 2%2F%28sin%282theta%29%29

= 2csc%282theta%29

So we could also say that
%28tan%28theta%29%29%2F%281+-+cos%5E2%28theta%29%29+=+2csc%282theta%29
is an identity.

You can use a graphing calculator to verify each identity.
Desmos is one good choice.
https://www.desmos.com/calculator/jwy2tauhgg
Notice how one curve overlaps the other perfectly. Click each curve on/off to help show the overlap (the curve should blink different colors).