if tan(x)= -2/5 and cos(x) is positive what is tan(2x)

Question

find using sum and difference identities

Answer 1

tan(2x) = 2sin(x)cos(x)/[cos²(x) - sin²(x)]

 

Let R² = X² + Y² where tan(x) = Y/X = -2/5,

and cos(x) = X/R > 0. So, R = √[ (2*2) + (5*5) ] =

√(4+25) = √29.

If tan(x) < 0 and cos(x) >0, then sin(x) must be -2/√29 or -2*√(29)/29 and cos(x) = 5*(29)/29.

Thus, tan(2x) = 2*(-2*√(29)/29)*(5*√(29)/29)/{ [(5*√(29)/29)]² - [-2*√(29)/29)]² }

= -20/{[25/29 - 4/29]}/29 = -20/21

Answer 2

Double angle formula would be more appropriate.

Any questons?