What is the identity being proved?

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In summary, we can simplify the given equation to $$\frac{\cot(A)\cos(A)}{\cot(A)+\cos(A)}=\frac{\cot(A)-\cos(A)}{\cot(A)\cos(A)}$$ and further simplify it to $$\frac{\cot(A)-\cos(A)}{\cot(A)\cos(A)}=\frac{\cos(A)}{1+\sin(A)}$$
  • #1
Silver Bolt
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$\frac{\cot\left({A}\right)\cos\left({A}\right)}{\cot\left({A}\right)+\cos\left({A}\right)}=\frac{\cot\left({A}\right)-\cos\left({A}\right)}{\cot\left({A}\right)\cos\left({A}\right)}$

$L.H.S=\frac{\cot\left({A}\right)\cos\left({A}\right)}{\cot\left({A}\right)+\cos\left({A}\right)}$

$=\frac{\frac{\cos\left({A}\right)}{\sin\left({A}\right)}\cos\left({A}\right)}{\frac{\cos\left({A}\right)}{\sin\left({A}\right)}+\cos\left({A}\right)}$

$=\frac{\frac{\cos^2\left({A}\right)}{\sin\left({A}\right)}}{\frac{\cos\left({A}\right)+\cos\left({A}\right)\sin\left({A}\right)}{\sin\left({A}\right)}}$

$=\frac{\cos^2\left({A}\right)}{\cos\left({A}\right)+\cos\left({A}\right)\sin\left({A}\right)}$

What should be done from here?
 
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  • #2
Silver Bolt said:
$\frac{\cot\left({A}\right)\cos\left({A}\right)}{\cot\left({A}\right)+\cos\left({A}\right)}=\frac{\cot\left({A}\right)-\cos\left({A}\right)}{\cot\left({A}\right)\cos\left({A}\right)}$

$L.H.S=\frac{\cot\left({A}\right)\cos\left({A}\right)}{\cot\left({A}\right)+\cos\left({A}\right)}$

$=\frac{\frac{\cos\left({A}\right)}{\sin\left({A}\right)}\cos\left({A}\right)}{\frac{\cos\left({A}\right)}{\sin\left({A}\right)}+\cos\left({A}\right)}$

$=\frac{\frac{\cos^2\left({A}\right)}{\sin\left({A}\right)}}{\frac{\cos\left({A}\right)+\cos\left({A}\right)\sin\left({A}\right)}{\sin\left({A}\right)}}$

$=\frac{\cos^2\left({A}\right)}{\cos\left({A}\right)+\cos\left({A}\right)\sin\left({A}\right)}$

What should be done from here?

to continue

$=\frac{\cos^2\left({A}\right)}{\cos\left({A}\right)( 1+\sin\left({A}\right))}$
$= \frac{\cos\left({A}\right)}{( 1+\sin\left({A}\right))}$
$= \frac{\cos\left({A}\right)(( 1-\sin\left({A}\right))}{( 1+\sin\left({A}\right))( 1-\sin\left({A}\right))}$ multiply by 1- sin for the case of 1 + sin so on
$= \frac{\cos\left({A}\right)(( 1-\sin\left({A}\right))}{( 1- \sin^2\left({A}\right))}$
$= \frac{\cos\left({A}\right)(( 1-\sin\left({A}\right))}{( \cos^2\left({A}\right))}$
$= \frac{(( 1-\sin\left({A}\right))}{ \cos\left({A}\right)}$
$= \frac{\cot(A) (( 1-\sin\left({A}\right))}{\cot\left({A}\right) \cos\left({A}\right)}$
Now for the numerator
$= \cot(A) (( 1-\sin\left({A}\right))$
$= \cot(A) - \cot(A)\sin\left({A}\right)$
$= \cot(A) - \cos(A)$

hence the result
 
  • #3
$$\frac{\cot(A)\cos(A)}{\cot(A)+\cos(A)}\cdot\frac{\tan(A)}{\tan(A)}=\frac{\cos(A)}{1+\sin(A)}$$

$$=\frac{\cos(A)(1-\sin(A))}{\cos^2(A)}=\frac{1-\sin(A)}{\cos(A)}\cdot\frac{\cot(A)}{\cot(A)}=\frac{\cot(A)-\cos(A)}{\cot(A)\cos(A)}$$
 

Related to What is the identity being proved?

1. How do I prove an identity?

To prove an identity, you must start with one side of the equation and manipulate it using algebraic properties and rules until it is equivalent to the other side of the equation. This shows that both sides are equal and the identity is proven.

2. What is the purpose of proving an identity?

The purpose of proving an identity is to show that two mathematical expressions are equivalent. This can help in simplifying expressions, solving equations, and verifying the correctness of mathematical statements.

3. Can an identity be proven false?

No, an identity is a statement that is true for all values of the variables involved. Therefore, if an identity is proven to be false, it means there is an error in the proof or an incorrect assumption was made.

4. Are there specific strategies for proving identities?

Yes, there are several strategies that can be used to prove identities, such as substitution, factoring, using trigonometric identities, and working with both sides of the equation separately. The best strategy to use will depend on the specific identity being proven.

5. Can identities be proven using only algebraic manipulations?

Yes, identities can be proven using only algebraic manipulations, but sometimes it may be easier to use other methods such as substitution or factoring. It is important to use the most efficient and effective method for each specific identity.

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