Question

7

Use the double-angle identities to find tan (2x) if cosx =
25
Enter an exact answer.
tan (2x) =
and sinx < 0.

Answer 1

Given the value of cos(x) and sin(x), we can substitute them into the identity and simplify to get the exact value of tan(2x) i.e. -48/169.

Step-by-step explanation:

To find tan(2x), we can use the double-angle identity for tangent:

tan(2x) = 2tan(x) / (1 - tan^2(x))

Given that cos(x) = 25/7 and sin(x) < 0, we can find sin(x) using the Pythagorean identity:

=> sin^2(x) + cos^2(x) = 1.

Plugging in the value of cos(x), we get:

sin^2(x) + (25/7)^2 = 1

Solving for sin(x), we find sin(x) = -24/7.

Now, we can substitute sin(x) and cos(x) into the double-angle identity for tangent:

tan(2x) = 2(-24/7) / (1 - (25/7)^2)

Simplifying, we get:

tan(2x) = -48/169