"Always, sometimes or never?"
"Is it always true, sometimes true or never true?"
These are questions that we should ask for everything in life, including mathematics. However, many students are accustomed to a fixed way of doing things, a fixed set of procedures to follow, a fixed type of solutions to problems. They will coast along until they encounter the following type of problem.
"What?!? I am used to seeing equations like 3 x + 1 = 7 which have one answer, and equations like x² = 9 which have 2 answers and inequalities like 2 x - 1 > 5 which has a range of answers, but not those above!?!"
Certainly so. This question is a little bit more challenging. But do not panic. Try to look for patterns. Observe that ^2-(\sin^{2}\theta})^2)
(\cos^{2}\theta-\sin^{2}\theta)})
(a-b)})
Remembering that 
For part (b) we proceed similarly.
We end up with a the statement like 
Now for the clincher!
Again we proceed as in parts (a) and (b) but note that both })
})
You may want to use the ASTC or "All Science Teachers are Crazy" mnemonic to help determine the quadrants where the cosine is positive or negative. For me, I prefer to consider the fact that the cosine is just the x-coordinate on the unit circle, so the cosine is positive on the right two quadrants (quadrants 1 and 4). We know that 
\frac{\pi}{2}})
\frac{\pi}{2}})
In conclusion
When solving inequalities or equations, we should consider the possibility that the it may always be true, or never be true, or sometimes true.
I hope you have learned something solving these challenging inequalities.
Image Credits
The first picture was derived from Pixabay, while the rest are all my own work.
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You might also want to read my recent articles:
Integral of 1/sqrt(e^2x+1) (Trigo Approach)
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"Singapore Math": The Ribbons Problem from PSLE 2017
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@tradersharpe
-- promoting sharp minds
img credz: pixabay.com
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Dang yeah, that stuff is what many students struggle with. 🐐
Mathematical terms & condition very tough.
If only I had you around when I was in school! @tradersharpe ;)
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