Question

Prove the identity by completing the table below, indicating the steps on the left and the reasoning on the right.

Answer 1

any attempts on the distributive property step yet?

A(B+C) = A*B + A*C

Answer 2

sec(x)*tan(x)+csc(x)*cot(x)

Answer 3

close but you have to distribute sec(x)csc(x) to both terms

Answer 4

sec(x)csc(x)*tan(x)+sec(x)csc(x)*cot(x)

Answer 5

good

then for the "apply the definitions step" you just re-write everything in terms of sin and cos using the appropriate definitions

be careful with parentheses

Answer 6

confused :/

Answer 7

sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)

Answer 8

so you'd replace everything with its appropriate definition

Answer 9

okay so sec(x)=1/cos(x)

Answer 10

good, keep going

you will apply the definitions to everything in the expression

sec(x)csc(x)*tan(x)+sec(x)csc(x)*cot(x)

Answer 11

sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)

Answer 12

like there are no numbers to put right, it will just be sin, cos, etc.

Answer 13

yes

Answer 14

okay so is :

sec(x) = 1/cos(x)
csc(x) = 1/sin(x)
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)

Answer 15

correct?

Answer 16

yes, but take the original expression and do the substitutions

Answer 17

sec(x)csc(x) = 1/cos(x)
csc(x) = 1/sin(x)
tan(x) = sin(x)/cos(x)
cot(x) = cos(x)/sin(x)

Answer 18

i did the first one, is that correct?

Answer 19

sec(x)csc(x) is not equal to 1/cos(x)

sec(x) is equal to 1/cos(x)
csc(x) is equal to 1/sin(x)

therefore sec(x)csc(x) = ?

Answer 20

sec(x)csc(x) =1/cos(x)+1/sin(x)

Answer 21

there's no plus sign it's just (1/cos(x))(1/sin(x))

Answer 22

now we have

(1/cos(x))(1/sin(x))*tan(x)+sec(x)csc(x)*cot(x)

keep going with the rest of the expression

Answer 23

tan(x) = 1/cos(x)/ 1/sin(x)

Answer 24

i guess so but there's a better way to write that, you can just write it as sin(x)/cos(x)

Answer 25

anyway now we have

(1/cos(x))(1/sin(x))*[sin(x)/cos(x)]+sec(x)csc(x)*cot(x)

keep going

Answer 26

cos(x)/sin(x)

Answer 27

good

(1/cos(x))(1/sin(x))*[sin(x)/cos(x)]+sec(x)csc(x)*cos(x)/sin(x)

keep going with the last sec and csc

Answer 28

Im having trouble with the sec and csc one

Answer 29

sec(x) is equal to 1/cos(x)
csc(x) is equal to 1/sin(x)

therefore sec(x)csc(x) = ?

Answer 30

sec(x)csc(x)=1/cos(x) (1/sin(x))

Answer 31

good so the entire expression becomes

(1/cos(x))(1/sin(x))*[sin(x)/cos(x)]+[1/cos(x)]*[(1/sin(x)]*cos(x)/sin(x)

Answer 32

now for the "simplify" step you just cross out expressions when applicable

for example, you have (1/sin(x))*[sin(x)] which cancels out to 1

Answer 33

sin(x)/cos(x)=tan(x)

Answer 34

hm, we don't want to re-write anything in terms of cot or tan because that would undo the work we did before

Answer 35

look for things you can cancel out between the numerator and denominator

Answer 36

oh okay..

Answer 37

then would it be wit cos

Answer 38

good, so when you cancel out the sin expression on the left and the cos expression on the right term what do you get?

Answer 39

the expression I am working with is: (1/cos(x))(1/sin(x))*[sin(x)/cos(x)]+[1/cos(x)]*[(1/sin(x)]*cos(x)/sin(x)

Answer 40

right

Answer 41

yes

Answer 42

csc^2(x)+sec^2(x)

Answer 43

kind of skipping a step there

after cancelling out you should get:

1/sin^2(x) + 1/cos^2(x)

then the next step asks you to re-write in terms of csc and sec to get

csc^2(x)+sec^2(x)

Answer 44

1/sin^2(x) + 1/cos^2(x)
csc^2(x)+sec^2(x)

Answer 45

so is this it for the simplifying step

Answer 46

notice how there's a simplifying step and a step where you apply the definitions of csc and sec

Answer 47

yes

Answer 48

this is the simplifying step

1/sin^2(x) + 1/cos^2(x)

this is the step where you apply csc and sec definitions

csc^2(x)+sec^2(x)

Answer 49

then for the applying pythagorean identities step you just replace csc^2 and sec^2 with their appropriate tan/cot values

Answer 50

csc^2(x)=1+cot^2(x)
sec^2(x)=tan^2(x)+1

Answer 51

good so adding them together = ?

Answer 52

adding both of the expressions together?

Answer 53

yes

you have csc^2(x)+sec^2(x) so you must add the expressions together

Answer 54

1+cot^2(x)+tan^2(x)+1

Answer 55

good

then for the last simplifying step just combine the like terms (1 and 1)

Answer 56

wow alot of steps

Answer 57

tan^2(x)+cot^2(x)+2

Answer 58

good, that's it

Answer 59

for the pyth identities right

Answer 60

1+cot^2(x)+tan^2(x)+1 is the pythagorean identities step

tan^2(x)+cot^2(x)+2 is the simplifying step

Answer 61

thankss!