Proof of Tangent Formula tan (α + β)

We will learn step-by-step the proof of tangent formula tan (α + β).

Prove that tan (α + β) = $$\frac{tan α + tan β}{1 - tan α tan β}$$

Proof: tan (α + β) = $$\frac{sin (α + β)}{cos (α + β)}$$

= $$\frac{sin α cos β + cos α sin β}{cos α cos β - sin α sin β}$$

= $$\frac{\frac{sin α cos β}{cos α cos β} + \frac{cos α sin β}{cos α cos β}}{\frac{cos α cos β}{cos α cos β} - \frac{sin α sin β}{cos α cos β}}$$, [dividing numerator and denominator by cos α cos β]

= $$\frac{tan α + tan β}{1 - tan α tan β}$$          Proved

Therefore, tan (α + β) = $$\frac{tan α + tan β}{1 - tan α tan β}$$

Solved examples using the proof of tangent formula tan (α + β):

1. Find the values of tan 75°

Solution:

tan 75° = tan ( 45° + 30°)

= tan 45° + tan 30°/1 - tan 45° tan 30°

= 1 + 1/√3/1 - (1 . 1/√3)

= √3 + 1/√3 - 1

= (√3+1)^2/(√3 - 1)( √3+1)

= (√3)^2 + 2 ∙ √3 + (1)^2/(3 - 1)

= 3 + 1 + 2 ∙ √3/(3 - 1)

= (4 + 2√3)/2

= 2 + √3

2. Prove that tan 50° = tan 40° + 2 tan 10°

Solution:

tan 50° = tan (40° + 10°)

⇒ tan 50° = tan 40° + tan 10/1 - tan 40° tan 10°

⇒ tan 50° (1 - tan 40° tan 10°) = tan 40° + tan 10°

⇒ tan 50° = tan 40° + tan 10° + tan 50° tan 40° tan 10°

⇒ tan 50° = tan 40° + tan 10° + 1 ∙ tan 10°, [since tan 50° = tan (90° - 40°) = cot 40° = 1/tan 40° ⇒ tan 50° tan 40° = 1]

⇒ tan 50° = tan 40° + 2 tan 10°              Proved

3. Prove that tan (45° + θ) = 1 + tan θ/1 - tan θ.

Solution:

L. H. S. = tan (45° + θ)

= tan 45° + tan θ /1 - tan 45° tan θ

= 1 + tan θ /1 - tan θ (Since we know that, tan 45° = 1)              Proved

3. Prove the identities:  tan 71° = cos 26° + sin 26°/cos 26° - sin 26°

Solution:

tan 71° = tan (45° + 26°)

= $$\frac{tan 45° + tan 26°}{1 - tan 45° tan 26° }$$

= 1 + tan 26°/1 - tan 26°

= [1 + sin 26°/cos 26°]/[1 - sin 26°/cos 26°]

= cos 26° + sin 26°/cos 26° - sin 26°              Proved

4. Show that tan 3x tan 2x tan x = tan 3x - tan 2x - tan x

Solution:

We know that 3x = 2x + x

Therefore, tan 3x = tan (2x + x) = $$\frac{tan 2x + tan x}{1 - tan 2x tan x}$$

⇒ tan 2x + tan x = tan 3x - tan 3x tan 2x tan x

⇒ tan 3x - tan 3x tan x = tan 3x - tan 2x - tan x              Proved

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