Question

(1+tan theta + sec theta ) ( 1+cot theta - cosec theta ) = ?
justify

Answer 1

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Answer 2

To find,

(1+tanθ + secθ ) ( 1+cotθ - cosecθ)

Solution,

We start with converting, tanθ, secθ, cotθ and cosecθ in the form of sinθ and cosθ.

• tanθ = sinθ/cosθ

• cotθ = cosθ/sinθ

• cosecθ = 1/sinθ

• secθ = 1/cosθ

Substituting the values in the given relation we get,

=  (1+tanθ + secθ ) ( 1+cotθ - cosecθ            

=  (1+sinθ/cosθ + 1/cosθ ) ( 1+cosθ/sinθ - 1/sinθ)

= ( 1+cosθ/sinθ - 1/sinθ) + sinθ/cosθ*( 1+cosθ/sinθ - 1/sinθ) + 1/cosθ*( 1+cosθ/sinθ - 1/sinθ)

= 1+cosθ/sinθ - 1/sinθ + sinθ/cosθ + 1 - 1/cosθ + 1/cosθ + 1/sinθ - 1/cosθ*sinθ

= 1 + cosθ/sinθ + sinθ/cosθ - 1/cosθ*sinθ

= 1 + ( $cos^{2}$θ + $sin^{2}$θ )/sinθ*cosθ - 1/cosθ*sinθ

{ $cos^{2}$θ + $sin^{2}$θ = 1 ]

=  1 + 1/cosθ*sinθ - 1/sinθ*cosθ

= 1

Therefore, the value of (1+tanθ + secθ ) ( 1+cotθ - cosecθ) is 1.