Questão resolvida - Demonstre as seguintes identidades trigonométricas_ a), b) e c) - Cálculo II - ESTÁCIO

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Tiago Lima - Instagram: @professor_disciplinas_exatas / WhatsAPP: (71) 9927-17449
 
• Demonstre as seguintes identidades trigonométricas:
 
a) cos x ⋅ tan x ⋅ cossec x = 1( ) ( ) ( )
 
Resolução:
 
Temos que;
 
tan x = e cossec x =( )
sen x
cos x
( )
( )
( )
1
sen x( )
 
Substituindo e simplificando fica;
 
cos x ⋅ tan x ⋅ cossec x = 1 cos x ⋅ ⋅ = 1 ⋅ = 1( ) ( ) ( ) → ( )
sen x
cos x
( )
( )
1
sen x( )
→
cos x
cos x
( )
( )
sen x
sen x
( )
( )
 
1 ⋅ 1 = 1 1 = 1→
 
 b) tan x ⋅ cossec x = 1 + tan x2( ) 2( ) 2( )
 
Já sabemos que;
tan x = e cossec x =( )
sen x
cos x
( )
( )
( )
1
sen x( )
Substituindo e simplificando fica;
tan x ⋅ cossec x = 1 + tan x ⋅ = 1 +2( ) 2( ) 2( ) →
sen x
cos x
( )
( )
2
1
sen x( )
2
sen x
cos x
( )
( )
2
 
⋅ = 1 + ⋅ =
sen x
cos x
2( )
2( )
1
sen x
2
2( )
sen x
cos x
2( )
2( )
→
1
cos x
2( )
sen x
sen x
2( )
2( )
cos x + sen x
cos x
2( ) 2( )
2( )
 
⋅ 1 = =
1
cos x
2( )
cos x + sen x
cos x
2( ) 2( )
2( )
→
sen x + cos x
cos x
2( ) 2( )
2( )
1
cos x
2( )
 
sen x + cos x = sen x + cos x = 12( ) 2( )
cos x
cos x
2( )
2( )
→
2( ) 2( )
 
 
(verdadeiro )
(verdadeiro)
 
c) tan x + 1 ⋅ 1 - tan x = 2 - sec x( ( ) ) ( ( )) 2( )
 
Resolução:
Temos que;
tan x = e sec x =( )
sen x
cos x
( )
( )
( )
1
cos x( )
 
Substituindo e simplificando fica;
tan x + 1 ⋅ tan x - 1 = 2 - sec x + 1 ⋅ - 1 = 2 -( ( ) ) ( ( ) ) 2( ) →
sen x
cos x
( )
( )
sen x
cos x
( )
( )
1
cos x( )
2
 
⋅ = 2 -→
sen x + cos x
cos x
( ) ( )
( )
cos x - sen x
cos x
( ) ( )
( )
1
cos x
2
2( )
 
⋅ - =→
sen x + cos x
cos x
( ) ( )
( )
sen x - cos x
cos x
( ) ( )
( )
2cos x - 1
cos x
2( )
2( )
 
- = 2cos x - 1→
sen x - cos x
cos x
2( ) 2( )
2( )
1
cos x
2( )
2( )
 
- sen x - cos x = 2cos x - 1→ 2( ) 2( )
cos x
cos x
2( )
2( )
2( )
 
cos x - sen x = 1 ⋅ 2cos x - 1→ 2( ) 2( ) 2( )
 
cos x - sen x = 2cos x - 1→ 2( ) 2( ) 2( )
 
cos x - sen x - 2cos x = - 1→ 2( ) 2( ) 2( )
 
-sen x - cos x = - 1 × -1→ 2( ) 2( ) ( )
 
sen x + cos x = 1→ 2( ) 2( )
 
 
(verdadeiro)