Verified answer

tan²θ = sin²θ / cos²θ

Hence

tan²θ = (1 - cos²θ)/cos²θ and

tan²θ = sin²θ / (1 - sin²θ)

Plugging in the value of tanθ,

16 cos²θ = 1 - cos²θ

16 (1 - sin²θ) = sin²θ

the value of cos²θ and sin²θ are

cos²θ = 1/17

sin² θ = 16/17

Given that tanθ < 0 and sinθ > 0, cosθ < 0

Therefore

cosθ = −√cos²θ =-1 / √17

sin θ = +√sin² θ = 4 / √17

secθ = 1 / cosθ = - √17

cscθ = 1 / sin θ = √17 / 4

cot θ = 1 / tanθ = -1 / 4

tan θ = sin θ / cos θ

therefore cos θ = -1/4

sin θ = +sqrt(15/16) or

sin θ = -sqrt(15/16)