how do u simply the trigonometric expression: 1+Sinu/Cosu+Cosu/1+Sinu?

Question

This is a trigonometric expression. Using trig. Identities and Equations

Answer 1

(1 + sinu)/cosu + cosu/(1 + sinu) ← assuming this is what you really mean.

= (1 + sinu)²/[cosu(1 + sinu)] + (cosu)²/[cosu(1 + sinu)]

= (1 + 2sin u + sin²u + cos²u)/[cosu(1 + sinu)]

= (2 + 2sinu)/[cosu(1 + sinu)] (Since sin²u + cos²u = 1)

= 2(1 + sinu)/[cosu(1 + sinu)]

= 2/cosu

= 2secu

Answer 2

First off, I'm going to use hypens to separate numerator and denominator, instead of using the backslash (/)

most useful trig identity here: cos^2(u) + sin^2(u) = 1

1. Find common denominator: use (cosu)(1 + sinu). We get:

(1+sinu)(1+sinu) + (cosu)(cosu)
----------------------- -------------------
cosu(1 + sinu)

2. multiply through on the numerator

1 + sinu + sinu + sin^2u + cos^u
--------------------------------------------
cosu(1+sinu)


3. Simplify

1 + 2sinu + sin^2u + cos^u
-----------------------------------
cosu(1+sinu)

4. See the sin^2u + cos^u in the last part of the numerator? The trig identity I listen for you earlier is useful here, because sin^2u + cos^u = 1.

2 + 2sinu
-----------------
cosu(1+sinu)

5. Factor out a 2 in the num.

2( 1 + sinu)
-------------------
cosu(1 + sinu)

6. the (1 + sinu) from num. and den. cancel, leaving:

2
-----------
cosu

7. 1/cosu = secu, but since there's a 2 in the num, we get:


2sec(u) as the answer.

Woohoo! :)

Answer 3

As stated 1+Sinu/Cosu+Cosu/1+Sinu only simplifies to

1+tanu+cosu+sinu

However, cosu / 1 seems pretty trivial. Perhaps you meant to say:

1+Sinu/Cosu+Cosu/(1+Sinu)

then for cosu/(1+sinu) multiply numerator and denominator by
1-sinu and you will get

cosu(1-sinu)/[(1+sinu)(1-sinu)]
=cosu(1-sinu)/(1-sin^2 u)
=cosu(1-sinu)/cos^2 u
=(1-sinu)/cosu
=secu - tanu

So

1+Sinu/Cosu+Cosu/(1+Sinu) = 1 + tanu + secu - tanu
= 1 + secu

Answer 4

Hi,

I will abbreviate sin u as s and cos u as c.

Problem loosing spacing

1 + s + c
. --- ---------- = original
. c 1 + s

1 + s + c (1 - s)
. --- -------------------- = multiply last by (1 - sin u)
. c (1 + s)(1 - s) top & bottom

1 + s + c (1 - s)
. --- -------------------- = 1 - sin^2 = cos^2
. c c^2

1 + s + c - cs
. --- --------------- = distribute numerator
. c c^2

1*c^2 + s*c + c - cs
-------- ------ --------------- = get common denominator c^2
. c^2 c*c c^2

. c^2 + s*c + c - cs
------------------------------------- = combine fractions
. c^2

. c^2 + c
---------------- = add like terms
. c^2

. c( c + 1 )
---------------- = factor top
. c^2

. c + 1
---------------- = cancel out c
. c

. c + 1
. ----- ----- = write as separate fractions
. c c


. 1 + sec u 1/cos u = sec u, its reciprocal

Answer 5

1 + tan u + cos u + sin u

i can reduce it to other things, but nothing seems simpler.