Question

If $0^0<x<{45}^0,$ then consider the following statements :
Assertion (A):
$\frac{1}{1+si{n}^{2}x}-\frac{1}{1+se{c}^{2}x}\ne \frac{1}{1+co{s}^{2}x}-\frac{1}{1+cose{c}^{2}x}$
Reason (R) : $sinx\ne cosx$
of these statements :

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true, but R is not the correct explanation of A.
• C. A is true, but R is false.
• D. A is false, but R is true.

Answer 1

The correct option is D A is false, but R is true.

Consider the first expression
$\frac{1}{1+si{n}^{2}x}-\frac{1}{1+se{c}^{2}x}$
$=\frac{1}{1+si{n}^{2}x}-\frac{co{s}^{2}x}{1+co{s}^{2}x}$
$=\frac{1+co{s}^{2}x-co{s}^{2}x-si{n}^{2}xco{s}^{2}x}{(1+si{n}^{2}x)(1+co{s}^{2}x)}$
$=\frac{1+co{s}^{2}x-co{s}^{2}x-si{n}^{2}xco{s}^{2}x}{1+\left(si{n}^{2}xco{s}^{2}x\right)+si{n}^{2}xco{s}^{2}x}$
$=\frac{1-si{n}^{2}xco{s}^{2}x}{2+si{n}^{2}xco{s}^{2}x}$ ...(i)
Let us now consider second expression
$\frac{1}{1+co{s}^{2}x}-\frac{si{n}^{2}x}{1+si{n}^{2}x}$
$=\frac{1+si{n}^{2}x-si{n}^{2}x-co{s}^{2}xsi{n}^{2}x}{(1+si{n}^{2}x)(1+co{s}^{2}x)}$
$=\frac{1-co{s}^{2}xsi{n}^{2}x}{1+\left(si{n}^{2}xco{s}^{2}x\right)+si{n}^{2}xco{s}^{2}x}$
$=\frac{1-co{s}^{2}xsi{n}^{2}x}{2+si{n}^{2}xco{s}^{2}x}$
$=\left(ii\right)$
However $sinx=cosxatx={45}^0$
$sinx$ is not equal to $cosx$ for $0<x<{45}^0$
Hence reason is true assertion is wrong