trignometry identites questions

Question

1.Why is it necessary to prove a trigonometric identity before using the identity?
2. If the identity 1+cot x=csc x is used in the proof of another identity, for what values of x is the proof valid?

 

 

Prove each identity. Show your steps 

1.sec(q)/tan(q)=csc (q)

2.sin(q)/1+sin(q )-  sin q/1-sin (q )= -2 tan^2 (q)

3.cos^2 (q)/1- sin(q)  - cos^2 q/1+sin(q)=2sin(q)

Answer 1

1)

 

We prove the identity to make sure the identity is a true statement.   To prove the identity, we simplify the longer side of the identity to see if it matches of the other side of the identity.

 

 

3)

 

I will do the first one for you.   Then try the next two on your own.

 

sec(q) = 1 / cos(q)       ,                tan(x) = sin(q)/cos(q)            and        csc(q) = 1/sin(q)

 

 

Plugging in these identities to prove the main identity,

 

(1 / cos(q)) / (sin(q)/cos(q)) =       

 

(1 / cos(q)) * (cos(q) / sin(q)) =

 

1 / sin(q) = csc(q)

 

 

Do you now understand how to prove identities?