SOLUTION: Prove the identity: {{{(tan(x)+1)^2=sec^2(x)(cos(x)+sin(x))^2}}}

Algebra ->  Trigonometry-basics -> SOLUTION: Prove the identity: {{{(tan(x)+1)^2=sec^2(x)(cos(x)+sin(x))^2}}}      Log On


   

Question 1199745: Prove the identity: %28tan%28x%29%2B1%29%5E2=sec%5E2%28x%29%28cos%28x%29%2Bsin%28x%29%29%5E2
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

%28tan%28x%29%2B1%29%5E2=sec%5E2%28x%29%28cos%28x%29%2Bsin%28x%29%29%5E2

tan%5E2%28x%29%2B2tan%28x%29%2B1=sec%5E2%28x%29%28cos%28x%29%2Bsin%28x%29%29%5E2












The identity is fully confirmed.

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Explanation:

To prove an identity, we alter one side only. The other side stays the same.
The steps above show the left hand side (LHS) transforming into the right hand side (RHS).
The RHS stays the same the entire time.

Since the RHS has cosine and sine, this is a hint to turn the tangent on the LHS into its equivalent form involving sine over cosine.
tan = sin/cos
which is of course the informal way to write this rule.

List of Trig Identities
https://tutorial.math.lamar.edu/pdf/Trig_Cheat_Sheet.pdf

Then I got each denominator the same at cos^2(x), and combined the fractions.

On the 6th step, I used the rule (a+b)^2 = a^2+2ab+b^2

On the last step, I used another trig identity of
sec = 1/cos