Q)  Prove the following trigonometry identity: (sin θ + cos θ)(cosec θ – sec θ) = cosec θ. sec θ – 2 tan θ

ICSE Specimen Question Paper (SQP)2025

Ans: 

Let’s start from LHS:

(sin θ + cos θ)(cosec θ – sec θ)

= sin θ cosec θ – sin θ sec θ + cos θ cosec θ – cos θ sec θ

= 1 – – 1

=

We know that sin ² θ + cos ² θ =  1

∴ cos ² θ = 1 – sin ² θ

by substituting the above in above equation, we get:

LHS =

=

=

=

= cosec θ sec θ – 2 tan θ

= sec θ cosec θ – 2 tan θ

= RHS ………. hence proved !

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