please solve my doubts

prove that

1) (1+ cot A +tan A)(sin A - cos A) / sec³ A -cosec³ A =sin² A cos² A

2) cosec A- sin A= a³ , sec A- cos A=b³ prove that : a²b²(a² +b²) =1

Question

please solve my doubts
prove that
1) (1+ cot A +tan A)(sin A - cos A) / sec3 A
-cosec3 A =sin2 A cos2 A
2) cosec A- sin A= a3 , sec A- cos A=b3
prove that : a2b2(a2
+b2) =1

Himanshu · Accepted Answer

$\frac{(1 + \cot A + \tan A) (\sin A - \cos A)}{\sec^{3} A - c \sec^{3} A} = \sin^{2} A \cos^{2} A L H S : \frac{(\frac{\sin A \cos A + \cos^{2} A + \sin^{2} A}{\sin A \cos A}) (\sin A - \cos A)}{\frac{\sin^{3} A - \cos^{3} A}{\sin^{3} A \cos^{3} A}} = \frac{(\sin^{2} A + \cos^{2} A + \sin A \cos A) (\sin A - \cos A) \sin^{2} A \cos^{2} A}{(\sin^{3} A - \cos^{3} A)} = \frac{(\sin^{3} A - \cos^{3} A) \sin^{2} A \cos^{2} A}{(\sin^{3} A - \cos^{3} A)} = \sin^{2} A \cos^{2} A$ $c \sec A - \sin A = a^{3}, \sec A - \cos A = b^{3} \Rightarrow \frac{\cos^{2} A}{\sin A} = a^{3}, \frac{\sin^{2} A}{\cos A} = b^{3} a = \frac{\cos^{\frac{2}{3}} A}{\sin^{\frac{1}{3}} A}, b = \frac{\sin^{\frac{2}{3}} A}{\cos^{\frac{1}{3}} A} a^{2} b^{2} (a^{2} + b^{2}) = 1 \frac{\cos^{2} A}{\sin A} (\frac{\cos^{\frac{2}{3}} A}{\sin^{\frac{1}{3}} A} \times \frac{\sin^{\frac{4}{3}} A}{\cos^{\frac{2}{3}} A}) + \frac{\sin^{2} A}{\cos A} (\frac{\cos^{\frac{4}{3}} A}{\sin^{\frac{2}{3}} A} \times \frac{\sin^{\frac{2}{3}} A}{\cos^{\frac{1}{3}} A}) = \cos^{2} A + \sin^{2} A = 1$

7

please solve my doubts prove that 1) (1+ cot A +tan A)(sin A - cos A) / sec3 A -cosec3 A =sin2 A cos2 A 2) cosec A- sin A= a3 , sec A- cos A=b3 prove that : a2b2(a2 +b2) =1

please solve my doubts

prove that

1) (1+ cot A +tan A)(sin A - cos A) / sec³ A -cosec³ A =sin² A cos² A

2) cosec A- sin A= a³ , sec A- cos A=b³ prove that : a²b²(a² +b²) =1