Verifying this Trigonometric Identity

In summary, the conversation discusses a trigonometry homework problem involving verifying trigonometric identities. The problem is to simplify (1-cos^2 (a))(1+cos^2(a)) = 2sin^2 (a) -sin^4 (a), and the conversation provides assistance on how to approach it. The solution involves using the Pythagorean identity and factoring to simplify the equation.
  • #1
Ivan92
201
3
Hey guys. How are you all doing? I'm helping my younger brother out with his trigonometry homework. He is dealing with verifying trigonometric identities. However, he has the problem that I am getting nowhere with. Hope you all can help. Thanks in advance. :)

Homework Statement



Verify (1-cos^2 (a))(1+cos^2(a)) = 2sin^2 (a) -sin^4 (a). I can't simplify the (1+cos^2(a)). Also can not tell if I can simplify the other side as well.

Homework Equations



sin^2 a + cos^2 a = 1

The Attempt at a Solution



So using the Pythagorean identity, I have been able to simplify this to:

(sin^2 (a))(1+cos^2) ) = 2sin^2 (a) -sin^4 (a).

I am just stuck in simplifying the part after sin^2 (a). Also can't seem to simplify the other side. Any assistance is awesome.
 
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  • #2
Start by factoring a sin^2 (a) from the sum on the right hand side. Then represent 2 as 1+1.

I think you'll see the rest after that.
 
  • #3
Ahh Awesome! Thank you StevenB. I appreciate it. My younger brother says thanks too. Haha! :)
 

FAQ: Verifying this Trigonometric Identity

What is the purpose of verifying a trigonometric identity?

The purpose of verifying a trigonometric identity is to prove that the given equation is true for all values of the variables involved, typically angles in radians or degrees. This is an important step in solving trigonometric equations and can also help in simplifying more complex expressions.

What are the common techniques used to verify trigonometric identities?

The most common techniques used to verify trigonometric identities include manipulating the equation using algebraic and trigonometric identities, using fundamental trigonometric identities, and using properties of even and odd functions.

What are some tips for verifying trigonometric identities?

Some tips for verifying trigonometric identities include starting with the more complex side of the equation, using substitution to simplify expressions, and simplifying both sides of the equation separately before trying to equate them.

What are some common mistakes to avoid when verifying trigonometric identities?

Some common mistakes to avoid when verifying trigonometric identities include forgetting to use parentheses when substituting values, making algebraic errors, and not simplifying both sides of the equation before equating them.

How can verifying trigonometric identities be helpful in real-world applications?

Verifying trigonometric identities can be helpful in real-world applications such as engineering, physics, and navigation. It allows for more accurate calculations and can help in solving real-world problems involving angles and trigonometric functions.

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