Solveeit Logo
Login

Question

Question: How do you simplify the expression \(\left( {1 - \cos \theta } \right)\left( {1 + \cos \theta } \rig...

How do you simplify the expression (1−cos⁡θ)(1+cos⁡θ)\left( {1 - \cos \theta } \right)\left( {1 + \cos \theta } \right) ?

Explanation

Solution

The given problem requires us to simplify the given trigonometric expression (1−cos⁡θ)(1+cos⁡θ)\left( {1 - \cos \theta } \right)\left( {1 + \cos \theta } \right). The question requires thorough knowledge of trigonometric functions, formulae and identities. We will use the algebraic identity (a−b)(a+b)=a2−b2\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2} to simplify the expression. Then, we use the trigonometry identity sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1 to get to the final answer.

Complete step by step solution:
In the given question, we are required to evaluate the value of (1−cos⁡θ)(1+cos⁡θ)\left( {1 - \cos \theta } \right)\left( {1 + \cos \theta } \right) using the basic concepts of trigonometry and identities.
First we simplify the given trigonometric expression using algebraic identity (a2−b2)=(a−b)(a+b)\left( {{a^2} - {b^2}} \right) = \left( {a - b} \right)\left( {a + b} \right). This identity simplifies the given trigonometric expression by expanding it as a product of two factors.
So, we have, (1−cos⁡θ)(1+cos⁡θ)\left( {1 - \cos \theta } \right)\left( {1 + \cos \theta } \right).
⇒(12−(cos⁡θ)2)\Rightarrow \left( {{1^2} - {{\left( {\cos \theta } \right)}^2}} \right)
⇒(1−cos⁡2θ)\Rightarrow \left( {1 - {{\cos }^2}\theta } \right)
Now, using the trigonometric identity sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1, we get
⇒1−(1−sin⁡2θ)\Rightarrow 1 - \left( {1 - {{\sin }^2}\theta } \right)
So, we have simplified the trigonometric expression a bit but it can be further simplified by opening the brackets. So, we get,
⇒1−(1−sin⁡2θ)\Rightarrow 1 - \left( {1 - {{\sin }^2}\theta } \right)
⇒sin⁡2θ\Rightarrow {\sin ^2}\theta
Therefore, we get the value of trigonometric expression (1−cos⁡θ)(1+cos⁡θ)\left( {1 - \cos \theta } \right)\left( {1 + \cos \theta } \right) as sin⁡2θ{\sin ^2}\theta .

Note:
Basic trigonometric identities include sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1, sec⁡2θ=tan⁡2θ+1{\sec ^2}\theta = {\tan ^2}\theta + 1 and cos⁡ec2θ=cot⁡2θ+1\cos e{c^2}\theta = {\cot ^2}\theta + 1. These identities are of vital importance for solving any question involving trigonometric functions and identities. All the trigonometric ratios can be converted into each other using the simple trigonometric identities listed above.