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Sign of Trigonometric Ratios in Different Quadrants

Prove the fol...

Question

Prove the following identity:
$(s i n A + s e c A)^{2} + (c o s A + c s c A)^{2} = (1 + s e c A . c s c A)^{2}$

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Solution

$L H S = (S i n A + \frac{1}{C o s A} + C o s A + \frac{1}{S i n A})$

$= {(\frac{(S i n A C o s A + 1)}{C o s A})}^{2} + {(\frac{(S i n A C o s A + 1)}{S i n A})}^{2}$

$= (\frac{(S i n A C o s A + 1)^{2}}{{C o s}^{2} A}) + (\frac{(S i n A C o s A + 1)^{2}}{{S i n}^{2} A})$

$= (S i n A C o s A + 1)^{2} * {\frac{1}{{C o s}^{2} A} + \frac{1}{{S i n}^{2} A}}$

$= {(\frac{S i n A C o s A}{S i n A C o s A} * \frac{1}{S i n A C o s A})}^{2}$

$= (1 + C o s e c A . S e c A)^{2}$

$= R H S$

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