Question

How do you verify this identity?

Answer 1

\[\tan \Theta * \sin \Theta + \cos \Theta = \sec \Theta?\]

Answer 2

What did you try?

Answer 3

Did you look at it in terms of sign and cosine? Perhaps try to get it into a fractional form and do the addition?

Answer 4

yes all you have to do is change tangent into sin/cos then multiply cos by cos/cos to get a common denominator moral of the story you will get sin^2+cos^2 over cos which wil give you sec... if that makes sense

Answer 5

You can start with the basic Pythagorean identity...
\[\large \sin^2(\theta) + \cos^2(\theta) = 1\]

Divide everything by \(\cos (\theta)\)
\[\large \frac{\sin^2(\theta)}{\cos(\theta)}+\frac{\cos^2(\theta)}{\cos(\theta)}=\frac1{\cos(\theta)}\]

Simplify...
\[\large \color{green}{\frac{\sin^2(\theta)}{\cos(\theta)}=\frac{\sin(\theta)\sin(\theta)}{\cos(\theta)}=\frac{\sin(\theta)}{\cos(\theta)}\times\sin(\theta)}=\color{blue}{\tan(\theta)\sin(\theta)}\]
Also...
\[\large\color{green}{\frac{\cos^2(\theta)}{\cos(\theta)}}=\color{blue}{\cos(\theta)}\]
And of course
\[\large \color{green}{\frac1{\cos(\theta}}=\color{blue}{\sec(\theta)}\]

And therefore, it follows that
\[\huge \tan(\theta)\sin(\theta)+\cos(\theta)=\sec(\theta)\]

Answer 6

Use:

\[\tan (\theta) = \frac{\sin(\theta)}{\cos(\theta)}\]\[\frac{\sin(\theta) \cdot \sin(\theta)}{\cos(\theta)} + \cos(\theta)\]

Now go ahead..

Answer 7

@terenzreignz You wrote a small math book there. Sweet!

Answer 8

I have that unhealthy tendency...
to be thorough, @e.mccormick :D

Answer 9

hey it is usually better to be that way especially when it comes to proving something

Answer 10

Thanks! @e.mccormick @jonsysibye @terenzreignz @waterineyes

I wasn't exactly sure how to solve it, i only had the sec = 1/cos part, and tan = sin/cos

Answer 11

@terenzreignz I have done that too. Turned one problem into a PDF with LaTeX so I could ue PGF to roperly graph the points to show what negative inverse slope was.

You are welcome toadytica305. have fun!

Answer 12

@e.mccormick Just saw you're a CS major.. Me too! :D

Answer 13

Fun fun. machines do exactly what we tell them... even if we tell them to do the wrong thing... hehe. Code is fun!