- #1
Sean1
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I cannot seem to prove the following identity
(1+sin(x))/(1-sin(x))=2tan^2(x)+1+2tan(x)sec(x)
Can you assist?
(1+sin(x))/(1-sin(x))=2tan^2(x)+1+2tan(x)sec(x)
Can you assist?
Prove Identity: (1+sin(x))/(1-sin(x))=2tan^2(x)+1+2tan(x)sec(x)
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In summary, the conversation is about trying to prove the identity (1+sin(x))/(1-sin(x))=2tan^2(x)+1+2tan(x)sec(x), and the speaker suggests using the conjugate to get rid of the denominator. The other person confirms that this approach is correct.
- #2
Jameson
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Hi Sean,
Let's start with the left-hand side.
\(\displaystyle \frac{1+\sin(x)}{1-\sin(x)}\).
We want this to turn into an expression without a fraction, so maybe we can try getting rid of the denominator somehow. When I see something in the form of $a-b$, I often try multiplying by the conjugate $a+b$.
\(\displaystyle \frac{1+\sin(x)}{1-\sin(x)} \left( \frac{1+\sin(x)}{1+\sin(x)} \right) \)
What do you get after trying this?
Let's start with the left-hand side.
\(\displaystyle \frac{1+\sin(x)}{1-\sin(x)}\).
We want this to turn into an expression without a fraction, so maybe we can try getting rid of the denominator somehow. When I see something in the form of $a-b$, I often try multiplying by the conjugate $a+b$.
\(\displaystyle \frac{1+\sin(x)}{1-\sin(x)} \left( \frac{1+\sin(x)}{1+\sin(x)} \right) \)
What do you get after trying this?
- #3
Sean1
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Thanks for getting me started.
This is my working. Can you confirm my approach is correct?
View attachment 4467
This is my working. Can you confirm my approach is correct?
View attachment 4467
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- #4
Jameson
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Looks good! :)
Related to Prove Identity: (1+sin(x))/(1-sin(x))=2tan^2(x)+1+2tan(x)sec(x)
1. How can you prove the identity (1+sin(x))/(1-sin(x)) = 2tan^2(x) + 1 + 2tan(x)sec(x)?
To prove this identity, we will use basic trigonometric identities and algebraic manipulation. We will start by rewriting the right side of the equation in terms of sine and cosine using the identity tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x). This will give us 2(sin(x)/cos(x))^2 + 1 + 2(sin(x)/cos(x))(1/cos(x)). Simplifying this expression will lead us to the left side of the equation, thus proving the identity.
2. Why is proving identities important in mathematics?
Proving identities is important because it helps us understand the relationships between different trigonometric functions and how they are related to each other. It also allows us to simplify complex expressions and solve equations more easily.
3. Can this identity be proven using a trigonometric identity proof?
Yes, this identity can be proven using a trigonometric identity proof. As mentioned earlier, we will use basic trigonometric identities to rewrite and simplify the expression on the right side of the equation to show that it is equal to the left side.
4. Are there any special cases where this identity does not hold true?
No, this identity holds true for all values of x. However, it is important to keep in mind that when x = π/2 + nπ (where n is an integer), the expression (1+sin(x))/(1-sin(x)) is undefined, so the identity would not hold true for these values.
5. How can I use this identity in real-world applications?
This identity can be used in various real-world applications, such as in physics and engineering, where trigonometric functions are commonly used. It can also be used in solving problems involving triangles and angles, as well as in graphing trigonometric functions.
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