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Maths-

General

Easy

Question

If $cos space left parenthesis A plus B plus C right parenthesis equals cos space A cos space B cos space C comma$ then $fraction numerator 8 sin space left parenthesis B plus C right parenthesis sin space left parenthesis C plus A right parenthesis sin space left parenthesis A plus B right parenthesis over denominator sin space 2 A sin space 2 B sin space 2 C end fraction$ =

1
-1
$\frac{1}{2}$
$- \frac{1}{2}$

hint Hint:

In the question first we will simplify the given trigonometric expression and the find the value of sin(A+B),sin(B+C), sin(C+A) using the trigonometric formula cos(A+B)=cosA.cosB -sinA.sinB. Then we will use the derived value to find the required result.

The correct answer is: -1

$cos space left parenthesis A plus B plus C right parenthesis equals cos space A cos space B cos space C rightwards double arrow cos A cos open parentheses B plus C close parentheses equals cos A. cos B. cos C rightwards double arrow cos A. cos open parentheses B plus C close parentheses minus sin A sin open parentheses B plus C close parentheses equals cos A. cos B. cos C rightwards double arrow cos A. cos open parentheses B plus C close parentheses minus cos A. cos B. cos C equals sin A sin open parentheses B plus C close parentheses rightwards double arrow cos A open square brackets cos open parentheses B plus C close parentheses minus cos B. cos C close square brackets equals sin A. sin open parentheses B plus C close parentheses rightwards double arrow cos A open square brackets cos B. cos C minus sin B. sin C minus cos B. cos C close square brackets equals sin A. sin open parentheses B plus C close parentheses rightwards double arrow sin open parentheses B plus C close parentheses equals fraction numerator negative cos A. sin B. sin C over denominator sin A end fraction S i m i l a r l y comma space sin open parentheses C plus A close parentheses equals fraction numerator negative cos B. sin C. sin A over denominator sin B end fraction sin open parentheses A plus B close parentheses equals fraction numerator negative cos C. sin A. sin B over denominator sin C end fraction sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals fraction numerator negative cos A. sin B. sin C over denominator sin A end fraction. fraction numerator negative cos B. sin C. sin A over denominator sin B end fraction. fraction numerator negative cos C. sin A. sin B over denominator sin C end fraction rightwards double arrow sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals negative sin A. sin B. sin C. cos A. cos B. cos C rightwards double arrow sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses equals fraction numerator negative sin 2 A. sin 2 B. sin 2 C over denominator 8 end fraction rightwards double arrow fraction numerator 8 sin open parentheses A plus B close parentheses. sin open parentheses B plus C close parentheses. sin open parentheses A plus C close parentheses over denominator sin 2 A. sin 2 B. sin 2 C end fraction equals negative 1$

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