Question

Simplify (1 − cos x)(1 + cos x)

Answer 1

Answer:

The expression is equivalent to $sin^{2}x$

Step-by-step explanation:

The first step is to multiply the two expressions between parentheses :

$(1-cosx).(1+cosx)=1+cosx-cosx-cos^{2}x=1-cos^{2}x$ (II)

There is a trigonometric identity that states :

$sin^{2}x+cos^{2}x=1$

Working with this expression :

$sin^{2}x+cos^{2}x=1$ ⇒

$sin^{2}x=1-cos^{2}x$ (I)

Using the equation (I) in (II) :

$(1-cosx).(1+cosx)=1-cos^{2}x=sin^{2}x$ ⇒

$(1-cosx).(1+cosx)=sin^{2}x$

Answer 2

You can use the Pythagorean Identities (first identity) to find out the simplified form of the given expression.

The simplified form of the given expression is  $sin^2(x)$

What are Pythagorean Identities?

$sin^2(\theta) + cos^2(\theta) = 1\\\\1 + tan^2(\theta) = sec^2(\theta)\\\\1 + cot^2(\theta) = csc^2(\theta)$

Using the above identity(first one) to simplify the given expression

The given expression is simplified as:

$(1 - cos(x))(1+cos(x)) = 1 - cos(x) + cos(x) -cos^2(x) = 1 - cos^2(x)\\\\ (1 - cos(x))(1+cos(x)) = sin^2(x) + cos^2(x) -cos^2(x) = sin^2(x)$

Thus,

The simplified form of the given expression is  $sin^2(x)$

Learn more about Pythagorean Identities here:

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