6. Trigonometry : Examples for Practice

Prove the following:2 Marks Each
$c o s e c θ \sqrt{1 - c o s^{2} θ} = 1$ $s e c θ \sqrt{1 - s i n^{2} θ} = 1$ $(c o s e c θ - c o t θ)^{2} = \frac{1 + c o s θ}{1 - c o s θ}$ ${⟮ s i n A + c o s e c A ⟯}^{2} + {⟮ c o s A + s e c A ⟯}^{2} = 7 + t a n^{2} A + c o t^{2} A$ $(c o s e c A - s i n A) (s e c A - c o s A) = \frac{1}{t a n A + c o t A}$ $\frac{s i n θ}{1 - c o s θ} = c o s e c θ + c o t θ$ $t a n θ - c o t θ = \frac{2 s i n^{2} θ - 1}{s i n θ c o s θ}$
Prove the following:3 Marks Each
$\sqrt{\frac{1 + c o s θ}{1 - c o s θ}} = c o s e c θ + c o t θ$ $\sqrt{\frac{1 - c o s θ}{1 + c o s θ}} = c o s e c θ - c o t θ$ $s e c A + t a n A = \sqrt{\frac{1 + s i n A}{1 - s i n A}}$ $s e c^{2} θ - c o s^{2} θ = s i n^{2} θ (s e c^{2} θ + 1)$ $c o s^{4} θ - c o s^{2} θ = s i n^{4} θ - s i n^{2} θ$