I'm in Pre-AP Pre-Cal and we have a chart of many of the Trig Identities but I'm lost. We are suppost to simplify expressions. That wasn't easy but I understand it occasionally but then today we were told to "Verify" Trig Identities. Lost. For example...

QUESTION: Cosx + Sinxtanx = SecX

I'll type my work so far but I start to go no where.. and I'm going to drop the x because it can't change and it's simpler to type without it
Cos + ((sin/1)(sin/cos)) = sec
Cos + Sin^2/cos = sec
cos + (1 - cos^2)/cos = sec
cos +(1/cos) - (cos^2/cos) = sec
cos + (1/cos) - (cos/1) = sec

... then I go now where... all that may be wrong, I'm not sure. I just need help understand. When they have two things next to each other to multiply them such as "sinxtanx".. I don't understand what to do what so ever. I just kind of go with it and hope I'm right.

I'm in Pre-AP Pre-Cal and we have a chart of many of the Trig Identities but I'm lost. We are suppost to simplify expressions. That wasn't easy but I understand it occasionally but then today we were told to "Verify" Trig Identities. Lost. For example...

QUESTION: Cosx + Sinxtanx = SecX

I'll type my work so far but i start to go no where..and I'm going to drop the x because it can't change and it's simplier to type without it
Cos + ((sin/1)(sin/cos)) = sec
Cos + Sin^2/cos = sec
cos + (1 - cos^2)/cos = sec
cos +(1/cos) - (cos^2/cos) = sec
cos + (1/cos) - (cos/1) = sec

...then i go now where....all that may be wrong, I'm not sure. I just need help understand. When they have two things next to eachother to multiply them such as "sinxtanx"..i don't understand what to do what so ever. i just kind of go with it and hope I'm right.

cos(x)+sin(x)tan(x)

cos(x)+sin(x)\cdot \frac{sin(x)}{cos(x)}

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

Multiply top and bottom of left side by cos(x):

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

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

The numerator is the famous useful identity that equal 1, so we have

\frac{1}{cos(x)}=sec(x)

Do not try to insert '1', but try other 'routes' before doing it. It's rare that the examiners or teachers ask you to insert a 1. It's more like eliminate two or more things to 1.

Tesia, you'll definitely want to trace through Galactus' approach, as it employs the Pythagorean identity. As Galactus points out, Sir Pythag is good fellow to make friends with, 'cause he's pretty darn helpful in a lot of situations.

But on a separate note, look how close you were in your own work-through: The left-hand side of your final line immediately simplifies to

\frac{1}{\text{cos}(x)} \ = \ \text{sec}(x)