Can Trigonometric Identities Be Proven Using Different Methods?

[physics kiddy]

Homework Statement

Prove that:
tan^2∅/tan∅ - 1 + cot^2∅/cot∅ - 1 = 1 + sec∅cosec∅

Homework Equations

The Attempt at a Solution

I have solved the question taking tan∅ = sin∅/cos∅.
But I want to solve it some other way.

 

[micromass]

What other way??
Why aren't you happy with your solution?

 

[SammyS]

Homework Statement

Prove that:
tan^2∅/tan∅ - 1 + cot^2∅/cot∅ - 1 = 1 + sec∅cosec∅

Homework Equations

The Attempt at a Solution

I have solved the question taking tan∅ = sin∅/cos∅.
But I want to solve it some other way.

What you wrote for the left hand side is literally (tan^2∅/tan∅) - 1 + (cot^2∅/cot∅) - 1, which is equivalent to tan∅ + cot∅ - 2 .

Assuming that you meant $\displaystyle \frac{\tan^2(\phi)}{\tan(\phi)-1}+\frac{\cot^2(\phi)}{\cot(\phi)-1}=1+\sec(\phi)\csc(\phi)\ ,$

yes there is another way. --- it's no better, but looks interesting enough. Even with it, eventually you will change tan to sin/cos or perhaps tan to sec/csc.

Change cot(ϕ) to 1/tan(ϕ) . Then multiply the numerator & denominator of the second fraction by -tan(ϕ) --- that will give you a common denominator. You can then get a difference of cubes in the numerator ...

 

[physics kiddy]

Thanks, I got the answer. But I have got one more question:

How to prove that slopes of perpendicular lines on graph paper have a product equal to -1 ?

 

[Infinitum]

Thanks, I got the answer. But I have got one more question:

How to prove that slopes of perpendicular lines on graph paper have a product equal to -1 ?

What is the angle(acute) between two lines of slopes say, m₁ and m₂?
When will they become perpendicular then?

 

[physics kiddy]

No idea !

 

[Infinitum]

Okay hmm, try drawing out two lines with a general angle θ between them. Say the angle the first line makes with the positive x-axis is A and the second line makes an angle B, now try finding a trignometrical relation between θ, A and B. (Hint: use the property of external angles)

 

[physics kiddy]

I have attached a pic. Tell me if it is like that.

 

[Infinitum]

The x-axis is not necessarily where the two lines meet. So you can draw them cutting the x-axis at different points, and such that they intersect somewhere arbitarily on the xy plane, for the sake of a more general result.