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

 Compound Angle






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.



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.




Share this page: What’s this?

Recent Articles

  1. Formation of Square and Rectangle | Construction of Square & Rectangle

    Jul 16, 25 02:45 AM

    Construction of a Square
    In formation of square and rectangle we will learn how to construct square and rectangle. Construction of a Square: We follow the method given below. Step I: We draw a line segment AB of the required…

    Read More

  2. Perimeter of a Figure | Perimeter of a Simple Closed Figure | Examples

    Jul 16, 25 02:33 AM

    Perimeter of a Figure
    Perimeter of a figure is explained here. Perimeter is the total length of the boundary of a closed figure. The perimeter of a simple closed figure is the sum of the measures of line-segments which hav…

    Read More

  3. Formation of Numbers | Smallest and Greatest Number| Number Formation

    Jul 15, 25 11:46 AM

    In formation of numbers we will learn the numbers having different numbers of digits. We know that: (i) Greatest number of one digit = 9,

    Read More

  4. 5th Grade Quadrilaterals | Square | Rectangle | Parallelogram |Rhombus

    Jul 15, 25 02:01 AM

    Square
    Quadrilaterals are known as four sided polygon.What is a quadrilateral? A closed figure made of our line segments is called a quadrilateral. For example:

    Read More

  5. 5th Grade Geometry Practice Test | Angle | Triangle | Circle |Free Ans

    Jul 14, 25 01:53 AM

    Name the Angles
    In 5th grade geometry practice test you will get different types of practice questions on lines, types of angle, triangles, properties of triangles, classification of triangles, construction of triang…

    Read More