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

Answer:

D. 1

Step-by-step explanation:

We have the expression, $\frac{\csc^{2}x\sec^{2}x}{\sec^{2}x+\csc^{2}x}$

We get, eliminating the cosecant function,

$\frac{\sec^{2}x}{\frac{\sec^{2}x}{\csc^{2}x}+1}$

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

i.e. $\frac{\sec^{2}x}{\frac{\sin^{2}x}{\cos^{2}x}+1}$

i.e. $\frac{1}{\cos^{2}x}\times \frac{\cos^{2}x}{\sin^{2}x+\cos^{2}x}$

Since, we know that, $\sin^{2}x+\cos^{2}x=1$

Thus,

$\frac{1}{\cos^{2}x}\times \frac{\cos^{2}x}{\sin^{2}x+\cos^{2}x}=1$

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

Hence, option D is correct.