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  4. In triangle A B C, the value of sin 2 A- cos B( cos A cos C+ cos B)- cos C( cos A cos B+ cos C) is

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Q. In triangle $A B C$, the value of $\sin ^{2} A-\cos B(\cos A \cos C+\cos B)-\cos C(\cos A$ $\cos B+\cos C)$ is

Trigonometric Functions

Solution:

$\sin ^{2} A-\cos B(\cos A \cos C+\cos B)-\cos C(\cos A \cos B+\cos C)$
$=\sin ^{2} A-\cos ^{2} B-\cos ^{2} C-2 \cos A \cos B \cos C$
$=\sin ^{2} A-\cos ^{2} B-\cos ^{2} C-\cos C(\cos (A+B)+\cos (A-B))$
$=\sin ^{2} A-\cos ^{2} B-\cos ^{2} C-\cos C(\cos (\pi-C)+\cos (A-B))$
$=\sin ^{2} A-\cos ^{2} B-\cos C \cos (A-B)$
$=\sin ^{2} A-\cos ^{2} B+\cos (A+B) \cos (A-B)$
$=\sin ^{2} A-\cos ^{2} B+\cos ^{2} A-\sin ^{2} B=1-1=0$