Question

If x = cosec A + cos A and y = cosec A - cos A then prove that
$( \frac{2}{x + y} ) {}^{2} + ( \frac{x - y}{2} ) {}^{2} = 1$
​

Answer 1

Step-by-step explanation:

Given:-

⇒x=cosecA+cosA

⇒y=cosecA−cosA

To prove:-

⇒(

x+y

2

)

2

+(

2

x−y

)

2

−1=0

∴x+y=cosecA+cosA+cosecA−cosA=2cosecA

x−y=cosecA+cosA−cosecA+cosA=2cosA

∴(

x+y

2

)

2

+(

2

x−y

)

2

−1=(

2cosecA

2

)

2

+(

2

2cosA

)

2

−1

⇒(sinA)

2

+(cosA)

2

−1

⇒sin

2

A+cos

2

A−1

⇒1−1

⇒0

Hence proved.