Processing math: 100%

Identities Involving Tangents and Cotangents

Identities involving tangents and cotangents of multiples or submultiples of the angles involved.

To prove the identities involving tangents and cotangents we use the following algorithm.

Step I: Express the sum of the two angles in terms of third angle by using the given relation.

Step II: Take tangent of the both sides.

Step III: expand the L.H.S. in step II by using the formula for the tangent of the compound angles

Step IV: Use cross multiplication in the expression obtain in step III.

Step V: Arrange the terms as per the requirement in the sum. If the identity involves cotangents, divide both sides of the identity obtained in step V by the tangents of all angles.

1. If A + B + C = Ο€, prove that, tan A + tan B + tan C = tan A tan B tan C.

Solution:

A + B + C = Ο€                                       

β‡’ A + B = Ο€ - C

Therefore, tan (A+ B) = tan (Ο€ - C)

β‡’ tanA+tanB1βˆ’tanAtanB = - tan C 

β‡’ tan A + tan B = - tan C + tan A tan B tan C

β‡’ tan A + tan B + tan C = tan A tan B tan C.                      Proved.

 

2. If A + B + C = Ο€2 prove that, cot A + cot B + cot C = cot A cot B cot C.

Solution:

A + B + C = Ο€2, [Since, A + B + C = Ο€2 β‡’ A + B = Ο€2 - C]

Therfore, cot (A + B) = cot (Ο€2 - C)

β‡’ cotAcotBβˆ’1cotA+cotB = tan C

β‡’ cotAcotBβˆ’1cotA+cotB = 1cotC

β‡’ cot A cot B cot C - cot C = cot A + cot B

β‡’ cot A + cot B + cot C = cot A cot B cot C.                      Proved.


3. If A, B and C are the angles of a triangle, prove that,
tan A2 tan B2+ tan B2 + tan C2 + tan C2 tan A2 = 1.

Solution:

 Since A, B, C are the angles of a triangle, hence, we have, A + B + C = Ο€
 A2 + B2 = Ο€2  - C2

β‡’ tan (A2 + B2) = tan (Ο€2  - C2)

β‡’ tan (A2 + B2) = cot C2

β‡’ tanA2+tanB21βˆ’tanA2βˆ™tanB2 = 1tanC2

β‡’ tan C2 (tan A2 + tan B2) = 1 - tan  A2  βˆ™ tan B2

β‡’ tan A2 tan B2 + tan B2 + tan C2 + tan C2 tan A2 = 1                  Proved.

● Conditional Trigonometric Identities






11 and 12 Grade Math

From Identities Involving Tangents and Cotangents 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. Volume of a Cuboid | Volume of Cuboid Formula | How to Find the Volume

    Jul 20, 25 12:58 PM

    Volume of Cuboid
    Cuboid is a solid box whose every surface is a rectangle of same area or different areas. A cuboid will have a length, breadth and height. Hence we can conclude that volume is 3 dimensional. To measur…

    Read More

  2. 5th Grade Volume | Units of Volume | Measurement of Volume|Cubic Units

    Jul 20, 25 10:22 AM

    Cubes in Cuboid
    Volume is the amount of space enclosed by an object or shape, how much 3-dimensional space (length, height, and width) it occupies. A flat shape like triangle, square and rectangle occupies surface on…

    Read More

  3. Worksheet on Area of a Square and Rectangle | Area of Squares & Rectan

    Jul 19, 25 05:00 AM

    Area and Perimeter of Square and Rectangle
    We will practice the questions given in the worksheet on area of a square and rectangle. We know the amount of surface that a plane figure covers is called its area. 1. Find the area of the square len…

    Read More

  4. Area of Rectangle Square and Triangle | Formulas| Area of Plane Shapes

    Jul 18, 25 10:38 AM

    Area of a Square of Side 1 cm
    Area of a closed plane figure is the amount of surface enclosed within its boundary. Look at the given figures. The shaded region of each figure denotes its area. The standard unit, generally used for…

    Read More

  5. What is Area in Maths? | Units to find Area | Conversion Table of Area

    Jul 17, 25 01:06 AM

    Concept of Area
    The amount of surface that a plane figure covers is called its area. It’s unit is square centimeters or square meters etc. A rectangle, a square, a triangle and a circle are all examples of closed pla…

    Read More