Proving Trig Identity: Secx - Tanxsinx = cosx Explained | One More Example

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SUMMARY

The discussion focuses on proving the trigonometric identity sec(x) - tan(x)sin(x) = cos(x). The solution involves manipulating the left side by expressing tan(x) as sin(x)/cos(x) and simplifying the expression to reach the conclusion. The key steps include rewriting the terms and factoring out common elements, ultimately leading to the identity being verified as true. The final simplification confirms that the left side equals cos(x).

PREREQUISITES
  • Understanding of trigonometric identities, specifically secant and tangent functions.
  • Knowledge of basic algebraic manipulation and factoring techniques.
  • Familiarity with the Pythagorean identity sin²(x) + cos²(x) = 1.
  • Ability to work with fractions and common denominators in trigonometric expressions.
NEXT STEPS
  • Study the derivation of the Pythagorean identities in trigonometry.
  • Learn how to manipulate trigonometric functions using algebraic techniques.
  • Explore additional examples of proving trigonometric identities for practice.
  • Investigate the relationship between secant, tangent, and sine functions in depth.
USEFUL FOR

Students learning trigonometry, educators teaching trigonometric identities, and anyone seeking to improve their skills in algebraic manipulation of trigonometric functions.

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Homework Statement


I'm finally starting to understand proving trig identities, but I have just one more that I can't seem to figure out.

secx - tanxsinx = cosx


Homework Equations


N/A


The Attempt at a Solution


Well first, I multiplied the tanx and sinx and came up with sin2x / cosx
Now I'm stuck. I'm trying to do 1/cosx - sin2x / cosx
Would the GCF be (cosx)(cosx) or just cosx, I'm drawing a blank here. I'm debating whether or not to change the sin2x to 1 - cos2x, but even if I do, I can't figure out how that would simplify to just cosx
 
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You're almost there\frac{1}{\cos x}-\frac{\sin^2x}{\cos x}=\frac{1}{\cos x}(1-\sin^2x). You should notice something very familiar now.
 
So would it simplify to cos2x / cosx, which would then simplify to cosx for my answer?
 
Yep it's that easy!
 

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