Question

Simplify the expression. Quantity cosecant of x to the power of two times secant of x to the power of two divided by quantity secant of x to the power of two plus cosecant of x to the power of two.

A. sin^2x
B. cos^2x
C. -1
D. 1

Answer 1

D. 1

Explanation:

We have the expression,
`(\csc^(2)x\sec^(2)x)/(\sec^(2)x+\csc^(2)x)`

We get, eliminating the cosecant function,

`(\sec^(2)x)/((\sec^(2)x)/(\csc^(2)x)+1)`

As, sinx is reciprocal of cosecx and cosx is reciprocal of secx,

i.e.
`(\sec^(2)x)/((\sin^(2)x)/(\cos^(2)x)+1)`

i.e.
`(1)/(\cos^(2)x)* (\cos^(2)x)/(\sin^(2)x+\cos^(2)x)`

Since, we know that,
`\sin^(2)x+\cos^(2)x=1`

Thus,

`(1)/(\cos^(2)x)* (\cos^(2)x)/(\sin^(2)x+\cos^(2)x)=1`

So, after simplifying, we get that the result is 1.

Hence, option D is correct.