Question

If q cos theta = p find tan theta - cot theta in terms of p and q.

Answer 1

Answer:

If q cos x = p, find the value of tan x - cot x, in terms of p and p.

cos x = p/q, so sin x = (q^2-p^2)^0.5/q. Then tan x = [(q^2-p^2)^0.5/q]/[p/q] =

[(q^2-p^2)^0.5]/p and cot x = p/[(q^2-p^2)^0.5].

Hence tan x - cot x =

[(q^2-p^2)^0.5]/p - p/[(q^2-p^2)^0.5]

= [(q^2-p^2) - p^2]/p[(q^2-p^2)^0.5]

= [q^2–2p^2]/p[(q^2-p^2)^0.5]

= [q^2–2p^2](q^2+p^2)^0.5]/p[(q^2-p^2)^0.5](q^2+p^2)^0.5]

= [q^2–2p^2](q^2+p^2)^0.5]/p[(q^2-p^2)]

Answer 2

Answer:

q² - 2p²/√(p²q² - p⁴)

Step-by-step explanation:

Cosθ = p/q

Sinθ = √1 - Cos²θ = √1 - p²/q² = 1/q[√q² - p²]

Tanθ = Sinθ/Cosθ = √q² - p² / p.

Cotθ = p/√q² - p²

Tanθ - Cotθ =  √q² - p² / p - p/√q² - p²

                   = (√q² - p² )² - p² / p(√q² - p²)

                   = q² - 2p²/√(p²q² - p⁴)