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Maths-

General

Easy

Question

If $tan space theta minus cot space theta equals a$ and $cos space theta minus sin space theta equals b$ then $open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared$

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hint Hint:

In the question we will simplify the given equations using some trigonometric identities.

The correct answer is: 4

$tan space theta minus cot space theta equals a rightwards double arrow fraction numerator sin theta over denominator cos theta end fraction minus fraction numerator cos theta over denominator sin theta end fraction equals a space space space space space space space rightwards double arrow fraction numerator sin squared theta minus cos squared theta over denominator cos theta sin theta end fraction equals a rightwards double arrow fraction numerator negative 2 cos 2 theta over denominator sin 2 theta end fraction equals a space space space space space space space space space open square brackets cos squared x minus sin squared x equals cos 2 x comma space 2 sin x space cos x space equals 2 sin 2 x close square brackets s q u a r i n g space b o t h space s i d e s comma rightwards double arrow fraction numerator 4 cos squared 2 theta over denominator sin squared 2 theta end fraction equals a squared a d d i n g space 4 space o n space b o t h space s i d e s comma rightwards double arrow fraction numerator 4 cos squared 2 theta over denominator sin squared 2 theta end fraction plus 4 equals a squared plus 4 rightwards double arrow fraction numerator 4 cos squared 2 theta plus 4 sin squared 2 theta over denominator sin squared 2 theta end fraction equals a squared plus 4 rightwards double arrow fraction numerator 4 over denominator sin squared 2 theta end fraction equals a squared plus 4........ e q u a t i o n space 1 space space space space space space space space space space space space space space open square brackets cos squared x plus sin squared x equals 1 close square brackets$
$cos space theta space minus sin theta space equals b s q u a r i n g space b o t h space s i d e s comma space rightwards double arrow open parentheses cos theta minus sin theta close parentheses squared equals b squared rightwards double arrow cos squared theta plus sin squared theta minus 2 sin theta cos theta equals b squared rightwards double arrow 1 minus sin 2 theta equals b squared space space space space space space open square brackets cos squared theta plus sin squared theta equals 1 comma space 2 sin theta cos theta equals sin 2 theta close square brackets rightwards double arrow negative sin 2 theta equals b squared minus 1 s q u a r i n g space b o t h space s i d e s comma rightwards double arrow sin squared 2 theta equals open parentheses b squared minus 1 close parentheses squared......... e q u a t i o n space 2$
on multiplying equation 1 and 2,
$open parentheses a squared plus 4 close parentheses open parentheses b squared minus 1 close parentheses squared equals fraction numerator 4 over denominator sin squared 2 theta end fraction cross times sin squared 2 theta equals 4$

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