Question

Consider the following statements :
1. $sin|x|+cos|x|$ is always positive.
2. $sin\left(x^2\right)+cos\left(x^2\right)$ is always positive.
Which of the above statements is/are correct ?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

The correct option is D Neither 1 nor 2

To prove that the statements are incorrect, it's enough to show an example, where the given statements are false.

As we can see fro the figure

$for\pi <x<\frac{3\pi }{2}$

$\sin \left|x\right|,\cos \left|x\right|<0$, hence $\sin \left|x\right|+\cos \left|x\right|<0$

The first statement is false

Similarly

$for\sqrt{\pi }<x<\sqrt{\frac{3\pi }{2}}$

$\sin x^2,\cos x^2<0$, hence $\sin x^2+\cos x^2<0$

Statement 2 is also false