prove that cos36°+sin36°/cos36°-sin36°= tan81°​

Step-by-step explanation:

AnswEr :

• To Prove :

\tan(9) = \dfrac{( \cos(36) - \sin(36) )}{( \cos(36) + \sin(36) )}tan(9)=

(cos(36)+sin(36))

(cos(36)−sin(36))

• Proof :

I'm taking LHS to Prove this Identity -

\Longrightarrow \sf \tan(9)⟹tan(9)

⠀⠀⠀⠀⋆ tan θ = sin θ / cos θ

\Longrightarrow \sf \dfrac{ \sin(9) }{ \cos(9) }⟹

cos(9)

sin(9)

⠀⠀⠀⠀⋆ we can write 9 = (45 - 36)

\Longrightarrow \sf \dfrac{ \sin(45 - 36) }{ \cos(45 - 36) }⟹

cos(45−36)

sin(45−36)

⠀⠀⠀⠀⋆ sin(A - B) = sinAcosB - cosAsinB

⠀⠀⠀⠀⋆ cos(A - B) = cosAcosB + sinAsinB

\Longrightarrow \sf \dfrac{ \sin(45) \cos(36) - \cos(45) \sin(36)}{ \cos(45) \cos(36) + \sin(45) \sin(36) }⟹

cos(45)cos(36)+sin(45)sin(36)

sin(45)cos(36)−cos(45)sin(36)

⠀⠀⠀⠀⋆ sin(45) = 1 /√2 = cos(45)

\Longrightarrow \sf \dfrac{ \dfrac{1}{ \sqrt{2} } \times \cos(36) - \dfrac{1}{ \sqrt{2} } \times \sin(36)}{ \dfrac{1}{ \sqrt{2} } \times \cos(36) + \dfrac{1}{ \sqrt{2} } \times \sin(36) }⟹

2

1

×cos(36)+

2

1

×sin(36)

2

1

×cos(36)−

2

1

×sin(36)

\Longrightarrow \sf \dfrac{ \cancel\dfrac{1}{ \sqrt{2} }( \cos(36) - \sin(36)) }{\cancel\dfrac{1}{ \sqrt{2} }( \cos(36) + \sin(36)) }⟹

2

1

(cos(36)+sin(36))

2

1

(cos(36)−sin(36))

\Longrightarrow \sf \dfrac{( \cos(36) - \sin(36)) }{( \cos(36) + \sin(36)) }⟹

(cos(36)+sin(36))

(cos(36)−sin(36))

⠀

\therefore \boxed{ \tan(9) = \dfrac{( \cos(36) - \sin(36) )}{( \cos(36) + \sin(36) )}}∴

tan(9)=

(cos(36)+sin(36))

(cos(36)−sin(36))

Answer:

To prove :

cos36° + sin36° / cos36° - sin36° = tan81°

Step-by-step explanation:

STEP 1 :

First, we consider the Left Hand Side (LHS),

STEP 2 :

Now, divide each term by cos36°, we get

STEP 3 :

By trigonometric identities, we know that   [tex]\frac{sin \alpha }{cos \alpha } = tan \alpha[/tex] , so we write

     ⇒          [tex]\frac{1 + tan 36^{0} }{1 - tan 36^{0} }[/tex]

since we rewrite the above equation as

     ⇒          [tex]\frac{1 + tan 36^{0} }{1 - 1. tan 36^{0} }[/tex]

STEP 4 :

We know that tan 45° = 1, the above equation is written as

     ⇒          [tex]\frac{tan 45^{0} + tan 36^{0} }{1 - tan 45^{0} .tan 36^{0} }[/tex]            -----> (1)

STEP 5 :

From the formula,  [tex]tan ( \alpha +\beta ) = \frac{tan \alpha + tan \beta }{1 - tan\alpha .tan \beta }[/tex]       -------> (2)

By comparing the equation (1) and (2), we write

                      α = 45° and β = 36°

STEP 6 :

Hence, tan (α + β) = tan (45° + 36°)

                             = tan 81°

                             = RHS

∴ LHS = RHS

Thus, cos36° + sin36° / cos36° - sin36° = tan81°

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