Math Help - I am in desperate need of help with these 5 identities

1. I am in desperate need of help with these 5 identities

Although I understand some, I've spent the last couple of hours trying to figure these out, and it seems hopeless. If you can explain step by step how one would go about solving these problems so that I can study and remember the methods I will greatly appreciate it!!

Well, here they are.

(tan x + cot x)^2 = sec^2x + csc^2x

(cosx / 1+ sinx) + (1+sinx / cosx) = 2secx

tan x * tan x/2 = sec x-1

1- tan^2x = ( 2tanx / tan2x )

( sin2x / 1+cos2x ) = tan x

Again, any help with the above problems would be great! I'm stressing over this very badly and it kills me that I do not understand it.

2. try tat table, n if u still din get em clearly then i ll post reply wid in 5hrs

$Each trigonometric function in terms of the other five. [4]Functionsincostancscseccotsinθ =cosθ =tanθ =cscθ =secθ =cotθ =" alt="Each trigonometric function in terms of the other five. [4]Functionsincostancscseccotsinθ =cosθ =tanθ =cscθ =secθ =cotθ =" />

List of trigonometric identities - Wikipedia, the free encyclopedia

3. Originally Posted by hi_mynameisdan
Although I understand some, I've spent the last couple of hours trying to figure these out, and it seems hopeless. If you can explain step by step how one would go about solving these problems so that I can study and remember the methods I will greatly appreciate it!!

Well, here they are.

(tan x + cot x)^2 = sec^2x + csc^2x

(cosx / 1+ sinx) + (1+sinx / cosx) = 2secx

tan x * tan x/2 = sec x-1

1- tan^2x = ( 2tanx / tan2x )

( sin2x / 1+cos2x ) = tan x

Again, any help with the above problems would be great! I'm stressing over this very badly and it kills me that I do not understand it.
$(\tan{x} + \cot{x})^2 = \sec^2{x} + \csc^2{x}$

$\tan^2{x} + 2\tan{x}\cot{x} + \cot^2{x} = \sec^2{x} + \csc^2{x}$

$\tan^2{x} + 2 + \cot^2{x} = \sec^2{x} + \csc^2{x}$

$(\tan^2{x} +1) + (\cot^2{x} + 1) = \sec^2{x} + \csc^2{x}$

$\sec^2{x} + \csc^2{x} = \sec^2{x} + \csc^2{x}$

4. Originally Posted by hi_mynameisdan
Although I understand some, I've spent the last couple of hours trying to figure these out, and it seems hopeless. If you can explain step by step how one would go about solving these problems so that I can study and remember the methods I will greatly appreciate it!!

Well, here they are.

(tan x + cot x)^2 = sec^2x + csc^2x

(cosx / 1+ sinx) + (1+sinx / cosx) = 2secx

tan x * tan x/2 = sec x-1

1- tan^2x = ( 2tanx / tan2x )

( sin2x / 1+cos2x ) = tan x

Again, any help with the above problems would be great! I'm stressing over this very badly and it kills me that I do not understand it.
Here is the first one:

$(\tan x + \cot x)^2 = \sec^2{x} + \csc^2{x}$

$(\tan x + \frac{1}{\tan x})^2 = \sec^2{x} + \csc^2{x}$

$\tan^2{x} + \frac{1}{\tan x} (\tan x) + \frac{1}{\tan x} (\tan x) + \frac{1}{\tan^2 {x} }= \sec^2{x} + \csc^2{x}$

$\tan^2{x} + 1 + 1 + cot^2 {x} = \sec^2{x} + \csc^2{x}$

$(1 + \tan^2{x}) + (cot^2 {x} + 1)= \sec^2{x} + \csc^2{x}$

$\sec^2{x} + \csc^2{x} = \sec^2{x} + \csc^2{x}$

There is no real secret to these kinds of equations, no failproof algorithm to apply. The best advice I can give you is to memorize the trigonometric identites -- really memorize them. That way you will be able to more easily see the relationships among different parts of the equation. In this particular example, recognizing the importance of the Pythagorean identities 1 + tan^2(x) = sec^2(x) and cot^2(x) + 1 = csc^2(x) is probably the most critical part.

5. Originally Posted by hi_mynameisdan
Although I understand some, I've spent the last couple of hours trying to figure these out, and it seems hopeless. If you can explain step by step how one would go about solving these problems so that I can study and remember the methods I will greatly appreciate it!!

Well, here they are.

(tan x + cot x)^2 = sec^2x + csc^2x

(cosx / 1+ sinx) + (1+sinx / cosx) = 2secx

tan x * tan x/2 = sec x-1

1- tan^2x = ( 2tanx / tan2x )

( sin2x / 1+cos2x ) = tan x

Again, any help with the above problems would be great! I'm stressing over this very badly and it kills me that I do not understand it.
The fourth one is super simple -- you just need to use the double angle identity for tan (2x) and then simplify.

6. Originally Posted by hi_mynameisdan
Although I understand some, I've spent the last couple of hours trying to figure these out, and it seems hopeless. If you can explain step by step how one would go about solving these problems so that I can study and remember the methods I will greatly appreciate it!!

Well, here they are.

(tan x + cot x)^2 = sec^2x + csc^2x

(cosx / 1+ sinx) + (1+sinx / cosx) = 2secx

tan x * tan x/2 = sec x-1

1- tan^2x = ( 2tanx / tan2x )

( sin2x / 1+cos2x ) = tan x

Again, any help with the above problems would be great! I'm stressing over this very badly and it kills me that I do not understand it.
For the fifth one, you need the double angle identites for sin and cos and the Pythagorean identity cos^2(x) + sin^2(x) = 1. If you need more help, let me know!

7. $\frac{\cos{x}}{1 + \sin{x}} + \frac{1 + \sin{x}}{\cos{x}}$

Find common den

$\frac{\cos^2{x} + (1 + \sin^2{x})^2}{(1 + \sin{x})(\cos{x})}$

$\frac{\cos^2{x} + 1 + 2\sin{x} + \sin^2{x}}{(1 + \sin{x})(\cos{x})}$

$\frac{(\cos^2{x} + \sin^2{x}) + 1 + 2\sin{x}}{(1 + \sin{x})(\cos{x})}$

$\frac{1 + 1 + 2\sin{x}}{(1 + \sin{x})(\cos{x})}$

$\frac{2 + 2\sin{x}}{(1 + \sin{x})(\cos{x})}$

$\frac{2(1 + \sin{x})}{(1 + \sin{x})(\cos{x})}$

$\frac{2}{\cos{x}}$

$2\sec{x}$