Prove : (1 cot^2x) (1-cosx) (1 cosx) = 1?

Question

Answer

( 1 - cos theta) ( 1 + cos theta ) (1 + cot square theta)

⇒ (1² - cos² theta) (1 + cot² theta) [ here, we just apply the formula a² - b² = (a+b) (a-b) ]

⇒ (1² - cos² theta) * cosec² theta [ we know that (1 + cot² theta) = cosec² theta]

⇒ sin² theta * cosec² theta [ we also know that (1² - cos² theta) = sin² theta ]

⇒ sin² theta * (1 / sin² theta) [ apply this formula (1 / sin² theta) = cosec² theta ]

⇒ 1

LHS =RHS
HENCE PROVED

Answer

Answer

Proof:

To prove the equation (1 cot^2x) (1-cosx) (1+cosx) = 1, we will simplify the left-hand side of the equation and show that it is equal to 1.

Simplifying the Left-hand Side:

Start by simplifying each term individually.

1. Simplify (1 cot^2x):
Recall that cot(x) = cos(x) / sin(x). Therefore, cot^2(x) = (cos^2(x) / sin^2(x)).
Multiplying by (1/sin^2(x)) on both the numerator and denominator, we get:
(cos^2(x) / sin^2(x)) * (1/sin^2(x)) = cos^2(x) / sin^4(x).

2. Simplify (1 - cos(x)):
Multiply (1 - cos(x)) by (1 + cos(x)) on both the numerator and denominator to get:
[(1 - cos(x)) / (1 + cos(x))] * [(1 + cos(x)) / (1 + cos(x))] = (1 - cos^2(x)) / (1 + cos^2(x)).
Using the identity sin^2(x) + cos^2(x) = 1, we can rewrite this as:
(sin^2(x) / (1 + cos^2(x))).

3. Simplify (1 + cos(x)):
No further simplification is needed for this term.

Combining the Simplified Terms:

Now, we can combine the simplified terms.

(1 cot^2x) (1 - cos(x)) (1 + cos(x)) = (cos^2(x) / sin^4(x)) * (sin^2(x) / (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x)) / (sin^4(x) * (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x)) / (sin^4(x) * (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x)) / (sin^4(x) * (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x)) / (sin^4(x) * (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x)) / (sin^4(x) * (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x)) / (sin^4(x) * (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x)) / (sin^4(x) * (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x)) / (sin^4(x) * (1 + cos^2(x))) * (1 + cos(x))
= (cos^2(x) * sin^2(x

Similar Class 10 Doubts

Prove it cot x . cot 2x - cot 2x . cot 3x - cot 3x . cot x = 1 ?

Prove that : tan theta/1-cot theta cot theta/1-tAn theta = 1 sec theta × coSec theta?

Prove that :(1+cotA +tan

Prove that 1/cosec theta -cot theta -1/sin theta=1/sin theta -1/cosec theta cot theta?

Prove If tan^3 theta/1+ tan^2 theta cot^3 theta/ 1+ cot^2 theta = sec theta ×cosec theta - 2 sin theta cos theta ?

Top Courses for Class 10View all

Science Class 10

Social Studies (SST) Class 10

Mathematics (Maths) Class 10

English Grammar Advanced

English Grammar Basic

Top Courses for Class 10

Top Courses for Class 10

Suggested Free Tests

Related Content

About Class 10

Class 10 Study Material

Class 10 All Subjects

NCERT Textbook Solutions

Other Classes

Important Links

Company