1. ## Simplifying Trigonometric Functions

I am having extreme troubles with this section. My book is no help and my teacher is clueless. Any help with these problems would be greatly helpful. Also if possible please include reasons/rules and show work to help me better understand and learn. Thanks!

1). tan(Pi/2-x)cscx/csc^2x

2). 1+tanx/1+cotx

3). (sec^2x+csc^2x)-(tan^2x+cot^2x)

4). sec^2u-tan^2u/cos^2v+sin^2v

2. hello bcavanaugh34
welcome to the forum

1). tan(Pi/2-x)cscx/csc^2x

assuming this is

$\displaystyle \frac {\tan\left({\frac{\pi}{2}-x}\right)\csc{x}} {\csc^2{x}} \rightarrow \frac {\tan\left({\frac{\pi}{2}-x}\right)} {\csc{x}} \rightarrow \frac{\cot{x}}{\csc{x}} \rightarrow \frac{\cos{x}}{\sin{x}}\times\frac{\sin{x}}{1} = \cos{x}$

hope this helps ..... still need more ?

3. More would be great. Thanks

4. ## #2

$\displaystyle \frac{1+\tan{x}}{1+cot{x}} \rightarrow \frac {\frac {\cos{x}+\sin{x}}{\cos{x}}} {\frac {\sin{x}+\cos{x}}{\sin{x}}} \rightarrow {\frac {\cos{x}+\sin{x}}{\cos{x}}} \times {\frac {\sin{x}}{\cos{x}+\sin{x}}} \rightarrow \frac{\sin{x}}{\cos{x}} =\tan{x}$

5. ## #3

$\displaystyle \left(\sec^2{\theta}+\csc^2{\theta}\right) -\left(\tan^2{\theta}+\cot^2{\theta}\right) \rightarrow \sec^2{\theta}+\csc^2{\theta}-\tan^2{\theta}-\cot^2{\theta}$
note that
$\displaystyle \tan^2{\theta}+1=\sec^2{\theta}$
and
$\displaystyle 1+\cot^2{\theta} = \csc^2{\theta}$
substituting these in:
$\displaystyle \tan^2{\theta}+1+1+\cot^2{\theta}-\tan^2{\theta}-\cot^2{\theta}=2$

6. ## #4

sec^2u-tan^2u/cos^2v+sin^2v

assuming this is

$\displaystyle \frac {\sec^2{u}-\tan^2{u}} {\cos^2{v}+\sin^2{v}} \rightarrow \frac{\sec^2{u}-(\sec^2{u}-1)}{1}=1$

7. Thank you so much but i am having trouble with the last section. It says: use the basic identities to change the expression to one involving only sines and cosines. Then simplify to a basic trigometric function.

1). (sinx)(tanx+cotx)

2). sinx-tan(x)cos(x)+ cos(pi/2-x)

3). sin(x)cos(x)tan(x)sec(x)csc(x)

8. Hello, bcavanaugh34!

The last two are almost too simple . . .

$\displaystyle 3)\;\; \left(\sec^2\!x+\csc^2\!x\right)-\left(\tan^2\!x+\cot^2\!x\right)$

$\displaystyle \text{We have: }\;\underbrace{\left(\sec^2\!x - \tan^2\!x\right)}_{\text{This is 1}} - \underbrace{\left(\csc^2\!x - \cot^2\!x\right)}_{\text{This is 1}} \;\;=\;\;1 + 1 \;\;=\;\;2$

$\displaystyle 4)\;\;\frac{\sec^2\!u-\tan^2\!u}{\cos^2\!v+\sin^2\!v}$

$\displaystyle \text{We have: }\;\frac{\overbrace{\sec^2\!u - \tan^2\!u}^{\text{This is 1}}}{\underbrace{\cos^2\!v + \sin^2\!v}_{\text{This is 1}}} \;\;=\;\;\frac{1}{1} \;\;=\;\;1$