# trigonometric equation

• Oct 3rd 2008, 10:16 AM
mi986
trigonometric equation
Hi, I have problem that asks, "Prove that the equations are identities."
I tried for hours to solve it, but I'm lost... I would really appreciate it if someone could help me on this.

cot A + tan A + 1 = (cot A)/(1-tan A) + (tan A)/(1-cot A)

The stipulation is that I can only use the trigonometric identities that I've learned so far..

Such as:
csc a = 1/sin a
sec a = 1/csc a
tan a = sin a/cos a
etc...

The pythgorean identities and Opposite angle formulas

So far I've tried switching it sines and cosines, tangent, and a few others but it never seems to come out right...

Thanks to anyone who can help.
• Oct 3rd 2008, 10:22 AM
Jhevon
Quote:

Originally Posted by mi986
Hi, I have problem that asks, "Prove that the equations are identities."
I tried for hours to solve it, but I'm lost... I would really appreciate it if someone could help me on this.

cot A + tan A + 1 = (cot A)/(1-tan A) + (tan A)/(1-cot A)

The stipulation is that I can only use the trigonometric identities that I've learned so far..

Such as:
csc a = 1/sin a
sec a = 1/csc a
tan a = sin a/cos a
etc...

The pythgorean identities and Opposite angle formulas

So far I've tried switching it sines and cosines, tangent, and a few others but it never seems to come out right...

Thanks to anyone who can help.

start by adding the fractions on the right hand side and simplify. now do you see how to get it?
• Oct 3rd 2008, 10:33 AM
mi986
That's one of the first things I did. :-s
• Oct 3rd 2008, 11:32 AM
Soroban
Hello, mi986!

Here's one way . .

Quote:

$\displaystyle \cot A + \tan A + 1 \:= \:\frac{\cot A}{1-\tan A} + \frac{\tan A}{1-\cot A}$

The right side is: .$\displaystyle \frac{\frac{1}{\tan A}}{1 - \tan A} + \frac{\tan A}{1 - \frac{1}{\tan }}$

Multiply each fraction by $\displaystyle \frac{\tan A}{\tan A}\!:\quad {\color{blue}\frac{\tan A}{\tan A}}\cdot\frac{\frac{1}{\tan A}}{1 - \tan A} + {\color{blue}\frac{\tan A}{\tan A}} \cdot\frac{\tan A}{1 - \frac{1}{\tan A}}$

. . $\displaystyle =\; \frac{1}{\tan A(1 - \tan A)} + \frac{\tan^2A}{\tan A - 1} \;=\;\frac{1}{\tan A(1 - \tan A)} - \frac{\tan^2\!A}{1-\tan A}$

Multiply the second fraction by $\displaystyle \frac{\tan A}{\tan A}\!:\quad\frac{1}{\tan A(1 - \tan A)} - \frac{\tan^3\!A}{\tan A(1 - \tan A)}$

. . and we have: .$\displaystyle \frac{1-\tan^3\!A}{\tan (1 - \tan A)}\;\;^{\leftarrow\;\text{difference of cubes}}$

Factor: .$\displaystyle \frac{(1-\tan A)(1 + \tan A + \tan^2\!A)}{\tan A(1 - \tan A)} \;=\;\frac{1 + \tan A + \tan^2\!A}{\tan A}$

. . $\displaystyle = \;\frac{1}{\tan A} + \frac{\tan A}{\tan A} + \frac{\tan^2\!A}{\tan A} \;=\; \cot A + 1 + \tan A$

• Oct 3rd 2008, 06:34 PM
mi986
Ooh! Thank you so much!