Prove the following trignometric identities:

$(1 - c o s^{2} A) c o s e c^{2} A = 1$

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Solution

By the trig identities

$c o s^{2} θ + s i n^{2} θ = 1$

=> $s i n^{2} θ = 1 - c o s^{2} θ$

and

$c o s e c^{2} θ$ = $\frac{1}{s i n^{2} θ}$

We have

$1 - c o s^{2} θ$ ( $c o s e c^{2} θ$ )

= $s i n^{2} θ$ ( $\frac{1}{s i n^{2} θ}$ )

= 1

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