What is the identity in p and q, if sin x+cos x=p and tan x+cot x=q?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll raise to square the 1st constraint sin x+cos x=p.

(sin x+cos x )^2= p^2

We'll expand the square:

(sin x)^2 + (cos x)^2 + 2sin x*cos x = p^2

We'll apply the Pythagorean identity for (sin x)^2 + (cos x)^2 = 1.

1 + 2sin x*cos x = p^2 (1)

We'll use the fact that tan x = sin x/cos x and cot x = cos x/sin x

We'll re-write the 2nd constraint:

sin x/cos x + cos x/sin x = q

[(sin x)^2 + (cos x)^2]/sin x*cos x = q

1/sin x*cos x = q

sin x*cos x = 1/q (2)

We'll substitute (2) in (1):

1 + 2/q= p^2

We'll multiply by q both sides:

q + 2 = q*p^2

We'll substract q both sides:

q*p^2 - q = 2

q(p^2 - 1) = 2

The relation between p and q, ifsin x+cos x=p and tan x+cot x=q, is: q(p^2 - 1) = 2.

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