Question

Prove the following trigonometric identities:

(secA - cosecA)(1 + tanA + cotA) = tanAsecA - cotAcosecA

Answer 1

To prove:  (secA - cosecA)(1 + tanA + cotA) = tanAsecA - cotAcosecA

Proof: Consider LHS,(secA-cosecA)(1+tanA+cotA)

We know, cosecA=1/sinA, secA=1/cosA, tanA=sinA/cosA,cotA=cosA/sinA

So,

(secA - cosecA)(1 + tanA + cotA) =\(\Big(\frac{1}{cosA}+\frac{1}{sinA}\Big)\)\(\Big(1+\frac{sinA}{cosA}+\frac{cosA}{sinA}\Big)\)

= \(\Big(\frac{sinA-cosA}{cosA sinA}\Big)\) \(\Big(\frac{sinA\,cosA+sin^2A-cos^2A}{cosA sinA}\Big)\)

Using the formula a³ - b³ = (a-b) (a²+b²+ab) we get,

= \(\frac{sin^3A-cos^3A}{sin^2A\,cos^2A}\)

R.H.S = tanAsecA -cotAcosecA

LHS = RHS 

Hence Proved