Question

Prove that sec*2 theeta + cosec*2 theeta =sec*2 theeta x cosec*2 theeta?

Answer 1

Proved in explanation...

Explanation:

`sec^2 theta+cosec^2theta` can be written as

`1/cos^2theta + 1/sin^2theta`

Taking lcm

`(sin^2theta+cos^2 theta)/(sin^2 theta*cos^2theta)`

`color (blue)("recall" sin^2 A+Cos^2 A=1)`

`1/(sin^2 theta* cos^2 theta)`

`1/sin^2theta *1/cos^2theta`

`cosec^2 theta×sec^2 theta`
Hence proved

Answer 2

See below.

Explanation:

I am reading this as:

`sec^2(theta)+csc^2(theta)=sec^2(theta)csc^2(theta)`

Identities:

1) `color(red)bb(sec^2(x)=1/cos^2(x))`

2) `color(red)bb(csc^2(x)=1/sin^2(x))`

3) `color(red)bb(sin^2x+cos^2x=1)`

`LHS`

`sec^2(theta)+csc^2(theta)`

Using identities 1 and 2:

`1/cos^2(theta)+1/sin^2(theta)`

Add fractions:

`(sin^2(theta)+cos^2(theta))/(cos^2(theta)sin^2(theta))`

Using identity 3

`1/(cos^2(theta)sin^2(theta))`

By identities 1 and 2:

`sec^2(theta)csc^2(theta)`

`LHS-=RHS`