Mathematics - Trigonometric Identities

Prove:

sinx + tanx = tanx (1 + cosx)

What I have so far:

LS:

= sinx + tanx
= sinx + (sinx / cosx)
= (sinx) (cosx) + sinx / cos
= tanx (cosx + sinx)

I don't know what to do now

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1. Your second last line needs brackets, and the following line makes no sense

Why not work on the right side?

RS = tanx(1+cosx)
=tanx + tanxcosx after expanding
= tanx + (sinx/cosx)cosx
= tanx + sinx
= LS

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2. My second last line:

(sinx) (cosx) + sinx
_________________________
cosx

When I did the right side, I get the same problem as the left side:

RS:

= tanx + (1 + cosx)
= (sinx / cosx) (1 + cosx)
= (1 + cosx) (cosx)
= (sinx) (1 + cosx)
_______________________
cosx
= (sinx) + (cosx)(sinx)
________________________
cosx

And ... my teacher wants everything in terms of cos and sin

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3. RS = tanx(1+cosx)
=tanx + tanxcosx after expanding
= tanx + (sinx/cosx)cosx
= tanx + sinx
= LS

For this line "=tanx + tanxcosx after expanding", how did you get the second tanx from and what has happened to the addition sign?

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4. from you second last line

(sinx) (cosx) + sinx
_________________________
cosx

= sinx[cosx + 1]/cosx , ....I took out a common factor

= tanx(cosx + 1)
= RS

You said "And ... my teacher wants everything in terms of cos and sin"

your teacher probably suggested changing everything to sines and cosines as you work your way through. Notice that is also what I did as I showed the steps.

I would never have restricted my students to use only a part of their trig knowledge, why not make use of all the relationships at your hand, picking the simplest way?

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5. secx/sinx - sinx/cosx = cotx

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