Question

12. Rewrite the expression in terms of the given function: (sec x + csc x)(sin x + cos x) - 2 - tan x; cotx O A. 2cot x B. cot x C. 2 + cotx D. 0

Answer 1

Answer: cot x

Explanation:

(sec x + csc x)(sin x + cos x) - 2 - tan x >simplify to sin/cos

`=((1)/(cos x ) +(1)/(sin x)) (sin x + cosx) -2-(sinx)/(cosx)` >find common denominator

for first parenthesis

`=((sinx+cosx)/(sin xcos x )) (sin x + cosx) -2-(sinx)/(cosx)` >Multiply the first 2

parenthesis

`=((sin^(2) x+2sin x cos x+cos^(2) x)/(sin xcos x )) -2-(sinx)/(cosx)` >Use identity sin²x+cos²x=1

`=((1 +2sin x cos x)/(sin xcos x )) -2-(sinx)/(cosx)` >Combine all fractions with

common denominator

`=(1 +2sin x cos x-2sinxcosx -sin^(2)x )/(sin xcos x )` >Simplify

`=(1 -sin^(2)x )/(sin xcos x )` >Use identity sin²x=1-cos²x

`=(1 -(1-cos^(2)x) )/(sin xcos x )` >Distribute negative

`=(1 -1+cos^(2)x )/(sin xcos x )` >simplify 1-1

`=(cos^(2)x )/(sin xcos x )` >simplify cos/cos

`=(cosx )/(sin x )` >Use identity cot=cos/sin

= cot x

Answer 2

Option B, cotangent x or cot x

Explanation:

First, I set up some shorthand based how each trig function operates in order to set up some conversion factors. You can also use trig identities if you are more familiar with those as the other answer suggests. That way is easier but it requires you to know the trig identities. If not, using the basic principles from angles of a right triangle can help:
Sine of x is the opposite leg over hypotenuse so we say S = O / H
Cosine of x is adjacent leg over hypotenuse so we say C = A / H
Tangent of x is opposite over hypotenuse so T = O / A
Cosecant of x is hypotenuse over opposite so csc = H / O
Secant of x is hypotenuse over adjacent so sec = H / A
Cotangent of x is adjacent over opposite so cot = A / O

For this first portion we are going to not think about the - 2 - tan x portion of the equation because we must FOIL the first part.

(sec x + csc x)(sin x + cos x)
FOIL stands for First, Outsides, Insides, and Lasts, marking what terms are multiply together in order to make an equation so:
Firsts: sec (sin x)
Outsides: sec (cos x)
Insides: csc (sin x)
Lasts csc (cos x)

So the new equation is:
sec (sin x) + sec (cos x) + csc (sin x) + csc (cos x)

Now we use our conversion factors to change each multiplication set:

`(H)/(A)((O)/(H)) + (H)/(A) ((A)/(H)) + (H)/(O)((O)/(H)) + (H)/(O)((A)/(H))`
Use your knowledge of multiplying fractions and how variables in the numerator and denominator can cancel each other out. You simplify to:

`(O)/(A) + 1 + 1 + (A)/(O)`
Now use the conversion factors again to convert what is left into trig functions. O / A is tan x. A / O is cot x.

tan x + 2 + cot x.

NOW, bring back the portion we neglected earlier, simplify and solve.
tan x + 2 + cot x - 2 - tan x
tan x - tan x + 2 - 2 + cot x
0 + 0 + cot x
0 + cot x
cot x, option B