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Question: How do you prove \[\left( {1 + \cos x} \right)\left( {1 - \cos x} \right) = {\sin ^2}x\] ?...

How do you prove (1+cos⁡x)(1−cos⁡x)=sin⁡2x\left( {1 + \cos x} \right)\left( {1 - \cos x} \right) = {\sin ^2}x ?

Explanation

Solution

Hint : The question is related to trigonometry, the sine, cosine, tangent, cosecant, secant, cotangent are trigonometry ratios and this can be abbreviated as sin, cos, tan, csc or cosec, sec, cot by using the algebraic formula (a2−b2)=(a+b)(a−b)\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right) and trigonometric identity sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1 . We get the required result.

Complete step-by-step answer :
Trigonometric ratios: Some ratios of the sides of a right angle triangle with respect to its acute angle called trigonometric ratios of the angle.
The ratios defined are abbreviated as sin x, cos x, tan x, csc x or cosec x, sec x, and cot x
Trigonometry having same identities that is trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables where both sides of the equality are defined. Geometrically, these are identities involving certain functions of one or more angles. The identities are :
sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1
1+tan⁡2x=sec⁡2x1 + {\tan ^2}x = {\sec ^2}x
1+cot⁡2x=csc⁡2x1 + {\cot ^2}x = {\csc ^2}x
Now we have to prove
⇒(1+cos⁡x)(1−cos⁡x)=sin⁡2x\Rightarrow \left( {1 + \cos x} \right)\left( {1 - \cos x} \right) = {\sin ^2}x
Let consider LHS
⇒(1+cos⁡x)(1−cos⁡x)\Rightarrow \left( {1 + \cos x} \right)\left( {1 - \cos x} \right)
Using the algebraic formula (a2−b2)=(a+b)(a−b)\left( {{a^2} - {b^2}} \right) = \left( {a + b} \right)\left( {a - b} \right) , then
⇒12−cos⁡2x\Rightarrow {1^2} - {\cos ^2}x
⇒1−cos⁡2x\Rightarrow 1 - {\cos ^2}x
Using the trigonometric identity sin⁡2x+cos⁡2x=1{\sin ^2}x + {\cos ^2}x = 1 , then the above equation becomes
⇒sin⁡2x\Rightarrow {\sin ^2}x
= RHS
Hence proved

Note : The trigonometry ratios are sine, cosine, tangent, cosecant, secant and cotangent. These are abbreviated as sin, cos, tan, cosec, sec and cot. To the trigonometry ratios we have trigonometry identities. These identities are used to simplify the trigonometric ratios. We can also solve this by the standard algebraic formula (a+b)(a−b)=a2−b2(a + b)(a - b) = {a^2} - {b^2} .