Q)  Prove that : tan θ /(1 – cot θ) + cot θ / (1 – tan θ) = 1 + sec θ + cosec θ

Ans:  Here, let’s start by simplifying the LHS in given equation:

LHS =

=

=

=

=

=

Now, we know that a^{3 –} b³ = (a – b) (a² + b² + a b) 

Hence, sin³ θ – cos³ θ  = (sin θ  – cos θ ) (sin² θ  + cos² θ  + sin θ  cos θ)

∵ sin² θ  + cos² θ  = 1

∴ sin³ θ – cos³ θ  = (sin θ  – cos θ ) (1 + sin θ  cos θ)

By substituting this value in nominator of LHS, we get:

LHS =

=

=

=

= 1 + sec θ cosec θ = RHS

Hence Proved !

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