Proving Trigonometric Identities: (sin φ+1-cos φ)/(sin φ+cos φ-1)

Another one is to use the trigonometric identity ##\tan x = \dfrac{\sin x}{\cos x}## and substitute for ##\tan \psi## in terms of sine and cosine. Then, you get an equation with only sines and cosines, which can be easily solved.In summary, the conversation revolved around finding an equivalent expression for the given equation, using trigonometric identities and simplification techniques. Three methods were suggested: expanding the quotients, using the identity for tangent, and substituting for tangent in terms of sine and cosine.
  • #1

chwala

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Homework Statement


Show that
## {(tan φ+sec φ-1)/(tan φ-sec φ+1)}≡ {(1+sin φ)/cos φ}##[/B]

Homework Equations

The Attempt at a Solution


## (sin φ+1-cos φ)/(sin φ+cos φ-1)##[/B]
 
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  • #2
chwala said:
## (sin φ+1-cos φ)/(sin φ+cos φ-1)##
How did you get this?
It is impossible to tell where you need help if you don't show the steps you took so far.
 
  • #3
To deal with expressions ##a-1## in a denominator, it is often useful to expand the quotient by ##a+1\,.##
 
  • #4
fresh_42 said:
To deal with expressions ##a-1## in a denominator, it is often useful to expand the quotient by ##a+1\,.##
That is what i did...let me look at my working again...
 
  • #5
No, you expanded the entire thing by ##\cos \varphi##. Now you can go ahead and write the denominator as ##a-1## and make ##a^2-1## out of it.
 
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  • #6
fresh_42 said:
No, you expanded the entire thing by ##\cos \varphi##. Now you can go ahead and write the denominator as ##a-1## and make ##a^2-1## out of it.
still not getting...how can you get ##a-1## in denominator?
look at my work now
##{(tan ψ+sec ψ-1)(tan ψ-sec ψ-1)}/{(tan ψ-sec ψ+1)(tan ψ-secψ-1)}##
let ## b= sec ψ-1##
we have
##{(tan ψ+b)(tanψ+b)}/{(tan^2ψ-b^2ψ)}##
after cancelling## (tan ψ+b)##
we get the original problem again!
 
  • #7
fresh_42 said:
No, you expanded the entire thing by ##\cos \varphi##. Now you can go ahead and write the denominator as ##a-1## and make ##a^2-1## out of it.
not really...how ## cos ψ?##
 
  • #8
anyway, let me try do it and post my solution, i believe i am capable...then you can post alternative way of doing it..
 
  • #9
The lefthand member uses tangents and secants. The righthand member uses sines and cosines. Try starting with definition of tangent and secant on the left side; and see what other simplifications and algebraic steps you can pick...
 
  • #10
I nailed it, i guess sometimes i am just too tired or not motivated. Here
## {(sin ψ+1-cosψ)/cosψ}.{(cos φ)/sin φ-1+cosφ)}##
##{(sin φ+1-cosφ)cos φ)/(sin φ-1+cosφ)cos φ)}##
##{sin ψcosψ+cosψ-cos^2ψ)/(sin φ-1+cosφ)cos φ)}##
##{sin ψcosψ+cosψ-1+sin^2ψ)/(sin φ-1+cosφ)cos φ)}##
##{(sin^2ψ-1+cosψ+sinψcosψ)/(sin φ-1+cosφ)cos φ)}##
##{(sin φ+1)(sinφ-1)+cosφ(1+sinφ)/(sin φ-1+cosφ)cos φ)}##
##{[(sin φ+1)][(sinφ-1+cosφ)]/(sin φ-1+cosφ)cos φ)}##
##(sin φ+1)/cosφ##
 
Last edited:
  • #11
Your equations would look much nicer if you used \sin and \cos in your tex expressions.
For example, ##\sin\psi## versus ##sin\psi##.
 
  • #12
There are 2 other methods to this...from my colleagues, i can share...
 
  • #13
I have already mentioned one: expand the quotients by ##\cos \varphi + \sin \varphi +1##.
 
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1. What are the basic trigonometry identities?

The basic trigonometry identities include the Pythagorean identities, reciprocal identities, quotient identities, and even-odd identities. These identities are used to simplify and solve trigonometric expressions.

2. How do I prove trigonometry identities?

To prove trigonometry identities, you must manipulate the expressions using algebraic techniques and apply the basic trigonometry identities. It is important to remember the fundamental trigonometric ratios and identities to successfully prove the identities.

3. What is the difference between a trigonometric equation and a trigonometric identity?

A trigonometric equation is an equation that involves trigonometric functions and can be solved for a specific value, while a trigonometric identity is an equation that holds true for all values of the variables. In other words, an identity is an equation that is always true, whereas an equation can only be true for certain values.

4. How can I use trigonometry identities to simplify expressions?

Trigonometry identities can be used to simplify expressions by replacing complicated expressions with simpler ones. By using the basic identities, you can manipulate the expressions and simplify them to a more manageable form.

5. Why are trigonometry identities important?

Trigonometry identities are important because they allow us to simplify and solve complex trigonometric expressions. They are also crucial for solving real-world problems involving angles and distances, such as in navigation, engineering, and physics.

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