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Maths-

General

Easy

Question

If $square root of 2 cos space A equals cos space B plus cos cubed space B text and end text square root of 2 sin text end text A equals sin space B minus sin cubed space B$ then sin (A-B) =

$\pm 1$
$\pm \frac{1}{2}$
$\pm \frac{1}{3}$
$\pm \frac{1}{4}$

hint Hint:

to find the value of the given equation we will use the formula Sin (A - B) = (Sin A * Cos B) - (Cos A * Sin B) and simplify the equation then using the given equations we will find the value of the required equations using the trigonometric identities.

The correct answer is: $plus-or-minus 1 third$

GIVEN-
$cos A = cos B + cos^{3} B..... equation 1$
$∴ cos A = 1/ \sqrt2 (cos B + cos^{3} B) ........ (Dividing both sides by \sqrt2)$
√2 sin A = sin B - sin³ B..... equation 2
∴ sin A = 1/√2 (sin B - sin³ B) ........ (Dividing both sides by √2)
TO FIND-
Sin (A - B)
SOLUTION-
We know that-
Sin (A - B) = (Sin A * Cos B) - (Cos A * Sin B)
∴ Sin (A - B) = [1/√2 (sin B - sin³ B) * cos B] - [1/√2 (cos B + cos³ B) * sin B] ...... (From Equations i & ii)
∴ Sin (A - B) = [1/√2 * (sin B Cos B - sin³ B Cos B)] - [1/√2 * (sin B cos B + cos³ B * sin B)]
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin³ B Cos B - (1/√2 sin B cos B + 1/√2 cos³ B * sin B)
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin³ B Cos B - 1/√2 sin B cos B - 1/√2 cos³ B * sin B
∴ Sin (A - B) = 1/√2 sin B Cos B - 1/√2 sin³ B Cos B - 1/√2 sin B cos B - 1/√2 cos³ B * sin B
∴ Sin (A - B) = -1/√2 sin³ B Cos B - 1/√2 cos³ B * sin B

∴ Sin (A - B) = -1/√2 sin B Cos B (sin² B +cos² B)
= -1/2√2 sin2B
Now squaring and adding equation 1 and 2 we get
$2 cos squared A plus 2 sin squared A equals cos squared B plus cos to the power of 6 B plus 2 cos to the power of 4 B plus sin squared B plus sin to the power of 6 B minus 2 sin to the power of 4 B rightwards double arrow 2 open parentheses cos squared A plus sin squared A close parentheses equals cos squared B plus sin squared B plus open parentheses cos squared B plus sin squared B close parentheses cubed minus 3 open parentheses cos squared B sin squared B close parentheses open parentheses cos squared B plus sin squared B close parentheses plus 2 open parentheses cos to the power of 4 B minus sin to the power of 4 B close parentheses rightwards double arrow 2 equals 1 plus 1 minus 3 over 4 sin squared 2 B plus 2 cos 2 B space space space space open square brackets 4 cos squared B sin squared B equals sin squared 2 B comma cos to the power of 4 B minus sin to the power of 4 B equals cos 2 B space close square brackets space space space open square brackets cos squared B plus sin squared B equals 1 close square brackets rightwards double arrow 0 equals negative 3 sin squared 2 B plus 8 cos 2 B rightwards double arrow 0 equals negative 3 open parentheses 1 minus cos squared 2 B close parentheses plus 8 cos 2 B space space space space open square brackets cos squared 2 B plus sin squared 2 B equals 1 close square brackets rightwards double arrow 0 equals 3 cos squared 2 B minus 3 plus 8 cos 2 B rightwards double arrow 3 cos squared 2 B plus 8 cos 2 B minus 3 equals 0 rightwards double arrow 3 cos squared 2 B plus 9 cos 2 B minus cos 2 B minus 3 equals 0 rightwards double arrow 3 cos 2 B open parentheses cos 2 B plus 3 close parentheses minus 1 open parentheses cos 2 B plus 3 close parentheses equals 0 rightwards double arrow open parentheses 3 cos 2 B minus 1 close parentheses open parentheses cos 2 B plus 3 close parentheses equals 0 rightwards double arrow cos 2 B equals 1 third space o r space cos 2 B equals negative 3 open parentheses n o t space p o s s i b l e close parentheses N o w comma space w e space k n o w space t h a t space open parentheses cos squared 2 B plus sin squared 2 B close parentheses equals 1 S o comma space 1 over 9 plus sin squared 2 B equals 1 rightwards double arrow sin squared 2 B equals 1 minus 1 over 9 equals 8 over 9 sin 2 B equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction t h e r e f o r e comma space sin open parentheses A minus B close parentheses equals fraction numerator negative sin 2 B over denominator 2 square root of 2 end fraction equals plus-or-minus fraction numerator 2 square root of 2 over denominator 3 end fraction cross times fraction numerator 1 over denominator 2 square root of 2 end fraction equals plus-or-minus 1 third$

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