Pls solve the ninth one.....

Pls solve the ninth one..... 2 tan2 A tan: A 2 21(B) (ix) A + stn A - I cos A + sin A + I = cosec A + sec A If x cos A + y Sin A = m and x sin A — y cos A = n. then prove that 2 2 If m a sec A + b tan A and

Dear Student,
Identities used-

\sin^{2} A + \cos^{2} A = 1 a^{2} - b^{2} = (a - b) (a + b) \frac{1}{\sin A} = \cos e c A \frac{1}{\cos A} = s e c A

LHS-

= \frac{1}{\cos A + \sin A - 1} + \frac{1}{\cos A + \sin A + 1} = \frac{\cos A + \sin A - 1 + \cos A + \sin A + 1}{(\cos A + \sin A + 1) (\cos A + \sin A - 1)} = \frac{2 \cos A + 2 \sin A}{{(\cos A + \sin A)}^{2} - {(1)}^{2}} = \frac{2 \cos A + 2 \sin A}{\cos^{2} A + \sin^{2} A + 2 \cos A \sin A - 1} = \frac{2 \cos A + 2 \sin A}{1 + 2 \cos A \sin A - 1} = \frac{2 \cos A}{2 \cos A \sin A} + \frac{2 \sin A}{2 \cos A \sin A} = \cos e c A + s e c A

Regards