Proofs Using Our
Identities
Reciprocal Relationships
1
sec  
cos 
1
csc  
sin 
1
cot  
tan 
Negative Relationships
csc( )   csc 
sin( )   sin 
y
y

-
x
-y
Negative Relationships
sec( )  sec 
cos( )  cos 
y

-
x
x
Negative Relationships
cot( )   cot 
tan( )   tan 
y
y

-
x
x
-y
x y r
2
y
2
2
P(x, y) or P(rcos, rsin)
r cos   r sin   r
2
r

2
2
2
2
y
x
x
r cos  r sin  r

 2
2
2
r
r
r
2
2
2
2
cos 2   sin 2   1
2
sin   cos   1
sin   cos   1
sin 2  cos 2 
1


2
2
sin  sin  sin 2 
sin 2  cos 2 
1


2
2
cos  cos  cos 2 
2
2
1  cot 2   csc2 
2
2
tan 2   1  sec2 
Pythagorean Relationships
sin   cos   1
2
2
1  cot 2   csc2 
tan   1  sec 
2
2
Start by simplifying the most
complicated side, here the left
side, and simplify until you get
the right side.
cot A  sec2 A

tan A
cos 2 A
1


2
2
sin A cos A
Prove:
cot A(1  tan 2 A)
2
 csc A
tan A
cot A  cot A  sec2 A  cot 2 A  sec2 A 
1

2
sin A
2
csc A
Prove:
sec x  sin x tan x  cos x
1
sin x
 sin x 

cos x
cos x
1  sin x

cos x
2
1
sin 2 x


cos x cos x
2
cos x

cos x
Prove:
Prove:
Prove:
tan x  cot x
 cot x
2
sec x
tan 2 
 1  sec
sec  1
ln 1  cos  ln 1  cos  2ln sin 
Prove:
tan 2 x
2

sin
x
2
1  tan x
Prove:
tan 
1  sec

 2csc
1  sec
tan 
Prove:
sin   cos  cos   sin 

 sec csc
sin 
cos 
Prove:
1  sin 2 x
2
2

sin
x
cos
x
2
1  cot x
Prove:
cot 2 x  cos2 x  sin 2 x  csc2 x
Prove:
(sec  tan )(csc  1)  cot 
Prove:
cos x  sin x  1  2cos x
Prove:
sec( x)  sec( x)sin 2 x  cos x
Prove:
1  csc
 cot   cos 
sec
4
4
2
Prove:
tan x  sin x
sec x

3
1  cos x
sin x
Prove:
sin x
1  cos x
2csc x 

1  cos x
sin x
Prove:
1  csc
sec x 
cot   cos 