Subscribe to our YouTube channel for the latest videos, updates, and tips.
We will learn step-by-step the proof of tangent formula tan (α + β).
Prove that tan (α + β) = tanα+tanβ1−tanαtanβ
Proof: tan (α + β) = sin(α+β)cos(α+β)
= sinαcosβ+cosαsinβcosαcosβ−sinαsinβ
= sinαcosβcosαcosβ+cosαsinβcosαcosβcosαcosβcosαcosβ−sinαsinβcosαcosβ, [dividing numerator and denominator by cos α cos β]
= tanα+tanβ1−tanαtanβ Proved
Therefore, tan (α + β) = tanα+tanβ1−tanαtanβ
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°)
= tan45°+tan26°1−tan45°tan26°
= 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) = tan2x+tanx1−tan2xtanx
⇒ 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
11 and 12 Grade Math
From Proof of Tangent Formula tan (α + β) to HOME PAGE
Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.
May 06, 25 01:23 AM
May 06, 25 12:01 AM
May 05, 25 01:27 AM
May 05, 25 01:05 AM
May 05, 25 12:23 AM
New! Comments
Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.