Math

# Prove that: cosecø + cotø / cosecø − cotø = 1 + 2cot²ø + 2cosec²ø cosø.​

Given :-

(cosecø + cotø)/(cosecø − cotø)

To find :-

Prove that:

(cosecø + cotø) / (cosecø − cotø )

= 1 + 2cot²ø + 2cosec²ø cosø.

Solution :-

On taking LHS

(cosecø + cotø)/(cosecø − cotø)

On multiplying both numerator and denominator with (cosecø + cotø) then

=> [(cosecø + cotø)/(cosecø − cotø)]× [(cosecø + cotø)/(cosecø + cotø)]

=> [(cosecø + cotø)(cosecø + cotø)] /[(cosecø + cotø)(cosecø − cotø)]

=> (cosecø + cotø)²/(cosec²ø − cot²ø)

Since , (a+b)(a-b) = a²-b²

=> (cosecø + cotø)²/1

Since Cosec² A - Cot² A = 1

=> (cosecø + cotø)²

=> cosec²ø + cot²ø+2cosecø.cotø

Since, (a+b)² = a²+2ab+b²

=> 1+cot²ø+cot²ø+2cosecø.cotø

Since,Cosec² A - Cot² A = 1

=> 1+2cot²ø+2cosecø.cotø

We know that

Cot A = Cos A / Sin A

=> 1+2cot²ø+2cosecø.(cosø/sinø)

=> 1+2cot²ø+2cosecø.(cosø× cosecø)

Since Cosec A = 1/Sin A

=> 1+2cot²ø+2cosec²ø.cosø

=> RHS

=> LHS = RHS

Hence, Proved.

If (cosecø + cotø) / (cosecø − cotø ) then

= 1 + 2cot²ø + 2cosec²ø cosø.

Used Algebraic Identities:-

→ (a+b)(a-b) = a²-b²

→ (a+b)² = a²+2ab+b²

Used Trigonometric Identities:-

→ Cosec² A - Cot² A = 1

Used Formulae:-

→ Cosec A = 1/Sin A

=> Cot A = Cos A / Sin A

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