Question

Assertion :If $0^0<x<{45}^0$, then consider the following statements
$\frac{1}{1+{\sin }^{2}x}-\frac{1}{1+{\sec }^{2}x}\ne \frac{1}{1+{\cos }^{2}x}-\frac{1}{1+{\text{cosec}}^{2}x}$ Reason: $\sin x\ne \cos x,x\ne \frac{\pi }{4},xϵ(0,\frac{\pi }{2})$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is D Assertion is incorrect but Reason is correct

$LHS$ $\to \frac{1}{1+{\sin }^{2}x}-\frac{1}{1+{Sec}^{2}x}$

$\to \frac{1}{1+{\sin }^{2}x}-\frac{1}{1+\frac{1}{{\cos }^{2}x}}$

$\to \frac{(1+{\cos }^{2}x)-{\cos }^{2}x(1+{\sin }^2\theta )}{(1+{\sin }^{2}x)(1+{\cos }^{2}x)}$

$\to \frac{1-{\cos }^{2}x{\sin }^{2}x}{(1+{\sin }^{2}x)(1+{\cos }^{2}x)}$

$RHS$ $\to \frac{1}{1+{\cos }^{2}x}-\frac{1}{1+{\text{cosec}}^{2}x}$

$\to \frac{1}{1+{\cos }^{2}x}-\frac{1}{1+\frac{1}{{\sin }^{2}x}}$

$\to \frac{(1+{\sin }^{2}x)-{\sin }^{2}x(1+{\cos }^{2}x)}{(1+{\cos }^{2}x)(1+{\sin }^{2}x)}=\frac{1-{\sin }^{2}x{\cos }^{2}x}{(1+{\sin }^{2}x)(1+{\cos }^{2}x)}$

Hence, assertion and reason, assertion is incorrect and reason is correct.