Simplifying Basic Trig: Use Identities

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SUMMARY

The discussion focuses on simplifying the expression \(\frac{\tan x + \tan x}{\csc^2 x + \sec^2 x}\) using basic trigonometric identities. Participants clarify that \(\tan(x)\) can be expressed as \(\frac{\sin(x)}{\cos(x)}\), \(\csc(x)\) as \(\frac{1}{\sin(x)}\), and \(\sec(x)\) as \(\frac{1}{\cos(x)}\). The simplification leads to the conclusion that the expression can be rewritten as \(\frac{2\tan x}{\csc^2 x + \sec^2 x}\), facilitating further simplification by factoring out \(\tan x\).

PREREQUISITES
  • Understanding of basic trigonometric identities
  • Familiarity with the definitions of \(\tan(x)\), \(\csc(x)\), and \(\sec(x)\)
  • Ability to manipulate algebraic fractions
  • Knowledge of sine and cosine functions
NEXT STEPS
  • Study the derivation and applications of trigonometric identities
  • Practice simplifying complex trigonometric expressions
  • Learn about the unit circle and its relationship to trigonometric functions
  • Explore advanced topics in trigonometry, such as inverse trigonometric functions
USEFUL FOR

Students studying trigonometry, educators teaching basic trigonometric identities, and anyone looking to enhance their understanding of trigonometric simplifications.

FAT
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okay so I need help with this. Directions say: "Use the basic identities to simplify to a basic trig function"

Tanx + Tanx
csc(squared)x + sec(squared)x


The lines are fractions. tanx/scs(squared)x for example. I need help with this thanks
 
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A lot of people define csc(x) as 1/sin(x) and sec(x) as 1/cos(x). Furthermore, some people even define tan(x) as sin(x)/cos(x). Does this help you?
 
FAT said:
okay so I need help with this. Directions say: "Use the basic identities to simplify to a basic trig function"

Tanx + Tanx
csc(squared)x + sec(squared)x


The lines are fractions. tanx/scs(squared)x for example. I need help with this thanks
It is not at all clear what you mean. Since you have "+" both above and below the fractionline I would have interpreted that as
[tex]\frac{tan x+ tan x}{csc^2 x+ sec^2 x}[/tex]
but then the numerator is obviously 2tan x.

Apparently you mean
[tex]\frac{tan x}{csc^2 x}+ \frac{tan x}{sec^2 x}[/tex]
As ZioX said, rewrite everything in terms of sin x and cos x and simplify.
(It might be easier to factor tan x out of that and concentrate on what's left!)
 

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