How do I prove this seemingly simple trigonometric identity

In summary, proving a trigonometric identity requires a logical and systematic approach, starting with simplifying both sides using known identities and manipulating one side to match the other. Useful identities include Pythagorean, double angle, and sum and difference identities. To check if an identity is true, you can substitute values or graph both sides. Algebraic manipulation, such as using the distributive property and factoring, is key in proving identities. Some tips for success include starting with the more complex side, breaking down the identity, using known identities, and being patient and thorough. A good understanding of trigonometric function properties and rules is also helpful.
  • #1
Ganesh Ujwal
56
0
Originally posted in a technical section, so missing the template
Mod note: Fixed the LaTeX.
##a=sinθ+sinϕ##

##b=tanθ+tanϕ##

##c=secθ+secϕ##Show that,

##8bc=a[4b^2 + (b^2-c^2)^2]##

I tried to solve this for hours and have gotten no-where. Here's what I've got so far :

##a= 2\sin(\frac{\theta+\phi}{2})\cos(\frac{\theta-\phi}{2}) ##

## b = \frac{2\sin(\theta+\phi)}{\cos(\theta+\phi)+\cos(\theta-\phi)}##

##c=\frac{2(\cos\theta+\cos\phi)}{\cos(\theta+\phi)+\cos(\theta-\phi)}##

##a^2 = \frac{\sin^2(\theta+\phi)[\cos(\theta+\phi)+1]}{\cos(\theta+\phi)+1}##

##\\cos(\theta-\phi)=\frac{ca}{b}-1##

##sin^2(\frac{\theta+\phi}{2})=\frac{2a^2b}{4(ca+b)}##
 
Last edited:
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  • #2
There's something wrong with your latex as it doesn't render at all.
 
  • #3
jedishrfu said:
There's something wrong with your latex as it doesn't render at all.
please fix it
 
  • #4
You need to edit your post to fix it. I don't have the authority to do that.

You're probably missing the tags for embedding latex in your post.
 
  • #5
how to embed latex? please show to me.
i am waiting
 
  • #6
Ganesh Ujwal said:
how to embed latex? please show to me.
\\ doesn't do anything. At the beginning and end of each line, put # # (no space between).
 
  • #7
alright then please anybody solve my problem, i am stuck.
 
  • #8
Did you try simply subbing a,b and c expressions into the right hand side to see what you get.

Also you didn't show us what identities you know that would be relevant to this problem.
 
  • #9
The ratio, a/b and a/c are very useful in this problem. Simplify your trig expressions for those a= , b= , and c= , and then using trig identities for sin(x+y) and cos(x+y), and cos(x-y), the algebra will simplify greatly. Do not immediately substitute those more complex trig expressions into your equation in a, b, and c.
 
  • #10
i tried like this:
\begin{align*}4\cos^2(\theta)\cos^2(\phi)b^2 &= 4\cos^2(\theta)\cos^2(\phi)(\tan(\theta) + \tan(\phi))^2 \\
&= 4(\sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi))^2 \\
&= 4\sin^2(\theta + \phi) \\
&= 16\sin^2((\theta + \phi)/2)\cos^2((\theta + \phi)/2) \\
&= 16(1 - \cos^2((\theta + \phi)/2))\cos^2((\theta + \phi)/2) \\
&= 16\cos^2((\theta + \phi)/2) - 16\cos^4((\theta + \phi)/2)\end{align*}
\begin{align*}c^2 - b^2 &= \sec^2\theta + \sec^2\phi + 2\sec(\theta)\sec(\phi) - (\tan^2\theta + \tan^2\phi + 2\tan(\theta)\tan(\phi)) \\
&= 2 + 2\sec(\theta)\sec(\phi) - 2\tan(\theta)\tan(\phi)\end{align*}
\begin{align*}\cos(\theta)\cos(\phi)(c^2 - b^2) &= 2(\cos(\theta)\cos(\phi) + 1 - \sin(\theta)\sin(\phi)) \\
&= 2(1 + \cos(\theta + \phi)) \\
&= 4\cos^2((\theta + \phi)/2)\end{align*}
\begin{align*}cos^2(\theta)\cos^2(\phi)(b^2 - c^2)^2 = 16\cos^4((\theta + \phi)/2\end{align*}
\begin{align*}cos^2(\theta)\cos^2(\phi)(4b^2 + (b^2 - c^2)^2) = 16\cos^2((\theta + \phi)/2\end{align*}
\begin{align*}8\cos^2(\theta)\cos^2(\phi)bc/a &= 8\cos^2(\theta)\cos^2(\phi)(\tan(\theta) + \tan(\phi))(\sec(\theta) + \sec(\phi))/(\sin(\theta) + \sin(\phi)) \\
&= 8(\sin(\theta)\cos(\phi) + \cos(\theta)\sin(\phi))(\cos(\theta) + \cos(\phi))/(\sin(\theta) + \sin(\phi)) \\
&= 8\sin(\theta + \phi)(\cos(\theta) + \cos(\phi))/(\sin(\theta) + \sin(\phi)) \\
&= \frac{8(2\sin((\theta + \phi)/2)\cos((\theta + \phi)/2))(2\cos((\theta + \phi)/2)\cos((\theta - \phi)/2))}{2\sin((\theta + \phi)/2)\cos((\theta - \phi)/2)} \\
&= 16\cos^2((\theta+\phi)/2)\end{align*}
now how should i proceed it?
 
  • #11
Ganesh Ujwal said:
alright then please anybody solve my problem, i am stuck.
The rules in this forum (under Homework Guidelines at https://www.physicsforums.com/threads/physics-forums-global-guidelines.414380/) don't permit solving your problem for you.
Giving Full Answers: On helping with questions: Any and all assistance given to homework assignments or textbook style exercises should be given only after the questioner has shown some effort in solving the problem. If no attempt is made then the questioner should be asked to provide one before any assistance is given. Under no circumstances should complete solutions be provided to a questioner, whether or not an attempt has been made.
 
  • #12
Ganesh Ujwal said:
now how should i proceed it?
Aren't you basically done? You have two expressions that equal ##16\cos^2\frac{\theta+\phi}2##. Set them equal to each other and simplify.
 

1. How do I approach proving a trigonometric identity?

Proving a trigonometric identity requires a logical and systematic approach. Start by simplifying both sides of the identity using known trigonometric identities. Then, try to manipulate one side to match the other side. If you get stuck, try working from both sides simultaneously.

2. What are some useful trigonometric identities to know?

Some useful identities include the Pythagorean identities (sin²x + cos²x = 1), double angle identities (sin2x = 2sinx cosx), and the sum and difference identities (sin(x+y) = sinxcosy + cosxsiny). It's important to have a good understanding of these basic identities before attempting to prove more complex ones.

3. How can I check if an identity is true or false?

You can substitute in specific values for the variables in the identity and see if both sides of the equation are equivalent. Another method is to graph both sides and see if they overlap. If both methods produce the same results, the identity is true.

4. How can I use algebra to prove a trigonometric identity?

Algebraic manipulation is a key tool in proving trigonometric identities. You can use algebraic properties such as the distributive property, combining like terms, and factoring to simplify one side of the identity and make it match the other side.

5. What are some tips for successfully proving a trigonometric identity?

Some tips for proving a trigonometric identity include: starting with the more complex side, breaking down the identity into smaller parts, using known identities to simplify, being patient and persistent, and checking your work carefully. It's also helpful to have a good understanding of the properties and rules of trigonometric functions.

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