given that sinx, tanx, and sin2x form geometric sequence, show cos^3x=1/2

can someone explain this to me, i have no idea how to solve it. i really done understand how to turn it into a geometric sequence

just think back to your Algebra II. Terms of a GP have a common ratio, so

tanx/sinx = sin2x/tanx
1/cosx = = 2sinx*cosx * cosx/sinx
...

To solve this problem, we need to understand the concept of a geometric sequence.

In a geometric sequence, each term is found by multiplying the previous term by a common ratio. Let's assume that sin(x), tan(x), and sin(2x) form a geometric sequence. This means that we can find any term in the sequence by multiplying the previous term by the same ratio.

Let's start with the first term of the sequence, which is sin(x). The next term is tan(x). To find the common ratio, we can divide the second term by the first term:

tan(x) / sin(x)

We know that tan(x) is equal to sin(x) / cos(x), so we can rewrite the common ratio as:

(sin(x) / cos(x)) / sin(x)

=sin(x) / [sin(x) * cos(x)]

Now, let's find the third term, which is sin(2x). To get this term, we multiply the second term (tan(x)) by the common ratio:

sin(x) / [sin(x) * cos(x)] * tan(x)

=sin(x) / [sin(x) * cos(x)] * [sin(x) / cos(x)]

=sin^2(x) / (sin(x) * cos(x))

Next, we need to simplify sin^2(x) / (sin(x) * cos(x)). Since sin^2(x) = (1 - cos^2(x)), we can substitute this expression into the equation:

(1 - cos^2(x)) / (sin(x) * cos(x))

Now, let's rewrite sin(x) * cos(x) as 1/2 * sin(2x) (double-angle identity):

(1 - cos^2(x)) / (1/2 * sin(2x))

To further simplify, we can multiply both the numerator and denominator by 2:

2 * (1 - cos^2(x)) / sin(2x)

Now, recall that the double angle identity for sin(2x) is 2 * sin(x) * cos(x). By substituting this into the equation, we get:

2 * (1 - cos^2(x)) / [2 * sin(x) * cos(x)]

This simplifies to:

(1 - cos^2(x)) / [sin(x) * cos(x)]

This expression is equal to cos^2(x) since sin^2(x) + cos^2(x) = 1. Therefore, we have:

cos^2(x) = cos^2(x)

Taking the cube root of both sides, we get:

cos(x) = 1/2

Finally, raising both sides to the power of 3, we obtain:

cos^3(x) = (1/2)^3 = 1/8

So, cos^3(x) is equal to 1/8.

Hence, we have shown that if sin(x), tan(x), and sin(2x) form a geometric sequence, then cos^3(x) is equal to 1/8.