Question

If m = CosecA - SinA and n = SecA – tanA
Prove that: (m^2n)^2÷3 + (mn^2)^2÷3} = 1​

Answer 1

Answer:

We have,

cosec{tex}\theta{/tex} - sin{tex}\theta{/tex} = m and sec{tex}\theta{/tex} - cos{tex}\theta{/tex} = n

{tex}\Rightarrow \quad \frac { 1 } { \sin \theta } - \sin \theta = m \text { and } \frac { 1 } { \cos \theta } - \cos \theta{/tex} = n

{tex}\Rightarrow \quad \frac { 1 - \sin ^ { 2 } \theta } { \sin \theta } = m \text { and } \frac { 1 - \cos ^ { 2 } \theta } { \cos \theta }{/tex} = n

{tex}\Rightarrow \quad \frac { \cos ^ { 2 } \theta } { \sin \theta } = m \text { and } \frac { \sin ^ { 2 } \theta } { \cos \theta }{/tex} = n

{tex}\therefore \quad \left( m ^ { 2 } n \right) ^ { 2 / 3 } + \left( m n ^ { 2 } \right) ^ { 2 / 3 } = \left( \frac { \cos ^ { 4 } \theta } { \sin ^ { 2 } \theta } \times \frac { \sin ^ { 2 } \theta } { \cos \theta } \right) ^ { 2 / 3 } + \left( \frac { \cos ^ { 2 } \theta } { \sin \theta } \times \frac { \sin ^ { 4 } \theta } { \cos ^ { 2 } \theta } \right) ^ { 2 / 3 }{/tex}

= (cos3{tex}\theta{/tex})2/3 + (sin3{tex}\theta{/tex})2/3 = cos2{tex}\theta{/tex} + sin2{tex}\theta{/tex} = 1

Hence, (m2n)2/3 + (mn2)2/3 = 1

Answer 2

Proved, $(m^{2}n)^{2/3} + (mn^{2})^{2/3} = 1$

Explanation:

$\csc\theta - \sin\theta = m$ and $\sec \theta - \cos\theta = n$

$\Rightarrow\frac{1}{\sin \theta }-\sin\theta = m$ and $\frac{1}{\cos\theta }-\cos\theta= n$  

$\Rightarrow \frac{1-\sin^{2}\theta }{\sin \theta }= m \text{and} \frac{1-\cos ^{2}\theta }{\cos \theta } = n$

$\Rightarrow \quad \frac{\cos^{2}\theta}{\sin \theta} = m \text { and } \frac { \sin ^ { 2 } \theta } { \cos \theta } = n$

$\therefore (m^{2}n)^{2 / 3} + ( m n ^ { 2 }) ^ { 2 / 3 } = ( \frac { \cos ^ { 4 } \theta } { \sin ^ { 2 } \theta } \times \frac { \sin ^ { 2 } \theta } { \cos \theta }) ^ { 2 / 3 } + ( \frac { \cos ^ { 2 } \theta } { \sin \theta } \times \frac { \sin ^ { 4 } \theta } { \cos ^ { 2 } \theta }) ^ { 2 / 3 }$

$=(\cos 3\theta)^{2/3} + (\sin 3\theta)^{2/3} =\cos2\theta+ \sin2\theta = 1$

Hence, $(m^{2}n)^{2/3} + (mn^{2})^{2/3} = 1$ .