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Maths-

General

Easy

Question

Let $P equals left curly bracket theta colon sin space theta minus cos space theta equals square root of 2 cos space theta right curly bracket text and end text Q equals left curly bracket theta colon sin space theta plus cos space theta equals square root of 2 sin space theta right curly bracket$ be two sets. Then

$P \subset Q and P \neq \emptyset$
$Q \subset\subset P$
$P \subset Q$
P=Q

hint Hint:

In the two sets $theta$ is derived from the given trigonometric equations in the respective sets. So, we will first simplify the equation then will get the relation between them.

The correct answer is: P=Q

Given two sets $P equals left curly bracket theta colon sin space theta minus cos space theta equals square root of 2 cos space theta right curly bracket text and end text Q equals left curly bracket theta colon sin space theta plus cos space theta equals square root of 2 sin space theta right curly bracket$
$F r o m space P comma space sin theta space minus cos theta equals square root of 2 cos theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space F r o m space Q comma space sin theta plus cos theta equals square root of 2 sin theta rightwards double arrow sin theta equals square root of 2 cos theta plus cos theta space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow cos theta equals square root of 2 sin theta minus sin theta rightwards double arrow sin theta equals cos theta open parentheses square root of 2 plus 1 close parentheses space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow cos theta equals sin theta open parentheses square root of 2 minus 1 close parentheses rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction equals square root of 2 plus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction equals fraction numerator 1 over denominator square root of 2 minus 1 end fraction cross times fraction numerator square root of 2 plus 1 over denominator square root of 2 plus 1 end fraction rightwards double arrow tan theta equals square root of 2 plus 1 space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space rightwards double arrow tan theta equals fraction numerator square root of 2 plus 1 over denominator 2 minus 1 end fraction equals square root of 2 plus 1 H e n c e comma space P equals Q$

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