Multiply and simplify. Check your result using a graphing calculator. (sinx-cosx)(sinx+cosx) sin

step 1 So we need to multiply and simplify this expression. So here this is A, this is B, this is A and

Multiply and simplify. Check your result using a graphing...

Key Concepts

Difference of Squares

The difference of squares is a fundamental algebraic identity which states that for any two expressions a and b, the product (a - b)(a + b) is equal to a² - b². In the context of trigonometric functions, this rule allows us to simplify products like (sin x - cos x)(sin x + cos x) into sin² x - cos² x, making further manipulations more straightforward.

Pythagorean Identity

The Pythagorean Identity is a key trigonometric relationship given by sin² x + cos² x = 1. This identity is essential when simplifying expressions involving powers of sine and cosine, as it allows one to substitute 1 - sin² x for cos² x (or vice versa), facilitating the combination and reduction of terms.

Algebraic Simplification

Algebraic simplification involves applying basic arithmetic and distribution principles to transform complex expressions into simpler or more recognizable forms. In this problem, after using the difference of squares and the Pythagorean identity, the expression is manipulated by distributing and combining like terms to arrive at a more concise result.

Transcript

00:01 So we need to multiply and simplify this expression.

00:04 So here this is a, this is b, this is a and again this is b.

00:08 This is a minus b, so we can apply formula a square minus b square.

00:15 So this will become sine square x minus cos square x times sine square x minus cost square x.

00:27 Now further simplification can be done by changing this cause square x.

00:31 Or sine square into sine or cos.

00:35 So this entire term can be written as cost 2x by half angle formula.

00:42 We give this sine square as it is and cost square can be written as 1 minus sine square x.

00:50 So this becomes cost 2x, sine square x minus 1 plus sine square x.

01:00 And now if we take sine square x common from this.

01:06 Term and this term we get cos 2x plus 1 minus 1 and what is 1 plus cos 2 theta 2 theta by 2 so sine square this becomes 2 cos square theta by 2 which is 2 x by 2 that will give us x minus 1 so now this is 2 times sine square x times cos square x and that can be written as sine x times cos x x whole square or individual squares and let's multiply two extra here and minus one so this entire term can be written as two sine x times cos x whole square by two minus one now what is two sine x x it is sine x x x x x x by 2 minus 1 so this is our simplified function we cannot simplify add it further.

02:11 Now we'll graph this function and this function and on graphing it, it looked like this.

02:20 Sorry, there is a calculation mistake here...

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