use the double angle formula to rewrite.
6cos^2-3
and
(cosx +sinx)(cosx-sinx)
cos ( 2 x ) = 2 cos ^ 2 ( x ) - 1
6 cos ^ 2 ( x ) - 3 =
3 [ 2 cos ^ 2 ( x ) - 1 ] =
3 sin ( 2 x )
second one:
(cosx +sinx)(cosx-sinx)
=(cosx +sinx)(cosx-sinx) * (cosx + sinx)/(cosx + sinx)
= (cos^2 x + 2sinxcosx + sin^2 x)/(cos^2 x - sin^2 x)
= (1 + sin 2x)/cos 2x
or
1/cos 2x + sin 2x/cos 2x
= sec 2x + tan 2x
To rewrite the expression 6cos^2(x) - 3 using the double angle formula, we first need to recall the formula for double angle in terms of cosine:
cos(2x) = 2cos^2(x) - 1
Now, let's go step by step:
1. Start with the expression: 6cos^2(x) - 3
2. Recognize that 6cos^2(x) can be rewritten as 3(2cos^2(x)).
3. Apply the double angle formula to 2cos^2(x) resulting in cos(2x).
4. Substitute cos(2x) in place of 2cos^2(x) - 1, giving us: 3(cos(2x) - 1) - 3.
Therefore, the expression 6cos^2(x) - 3 can be rewritten as 3(cos(2x) - 1) - 3.
Moving on to the second expression:
To expand (cos(x) + sin(x))(cos(x) - sin(x)), we can use the formula for multiplying binomials:
(a + b)(a - b) = a^2 - b^2
1. Start with the expression: (cos(x) + sin(x))(cos(x) - sin(x))
2. Notice that we have the form (a + b)(a - b), where a = cos(x) and b = sin(x).
3. Using the formula for multiplying binomials, substitute a^2 = cos^2(x) and b^2 = sin^2(x), resulting in cos^2(x) - sin^2(x).
Therefore, the expression (cos(x) + sin(x))(cos(x) - sin(x)) simplifies to cos^2(x) - sin^2(x).