If $m = a s e c A + b t a n A a n d n = a t a n A + b s e c A, t h e n p r o v e t h a t : m^{2} - n^{2} = a^{2} - b^{2}$

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Solution

m²-n²=(m+n)(m-n)
= $((a s e c A + b t a n A) + (a t a n A + b s e c A)) \times ((a s e c A + b t a n A) - (a t a n A + b s e c A))$
= $((a + b) s e c A + (a + b) t a n A) \times ((a - b) s e c A - (a - b) t a n A)$
= $(a + b) (s e c A + t a n A) \times (a - b) (s e c A - t a n A)$
= $(a + b) (a - b) (s e c A + t a n A) (s e c A - t a n A)$
= $(a ² - b ²) (s e c ² A - t a n ² A)$
= $(a ² - b ²) (1 + t a n ² A - t a n ² A)$
= $a ² - b ²$

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