a.
\(\sin47^0+\sin61^0-\left(\sin11^0+\sin25^0\right)\)
\(=2\sin54^0\cos7^0-2\sin18^0\cos7^0\)
\(=2\cos7^0\left(\sin54^0-\sin18^0\right)\)
\(=4\cos7^0.\cos36^0.\sin18^0\)
b.
\(\cos5x+\cos11x+3\left(\cos7x+\cos9x\right)\)
\(=2\cos8x.\cos3x+6\cos8x.\cos x\)
\(=2\cos8x\left(\cos3x+3\cos x\right)\)
\(=2\cos8x\left(4\cos^3x-3\cos x+3\cos x\right)\)
\(=8\cos8x.\cos^3x\)
c.
\(1-2\cos^2\dfrac{3x}{2}+\sin x-\cos5x-\sin7x\)
\(=-\cos3x-\cos5x+\sin x-\sin7x\)
\(=-2\cos4x.\cos x-2\cos4x.\sin3x\)
\(=-2\cos4x\left(\cos x+\sin3x\right)\)
d.
\(\sin3x+\sin x-\sin2x+2\left(1-\cos x\right)\cos x\)
\(=2\sin2x.\cos x-\sin2x+2\cos x-2\cos^2x\)
\(=2\cos x\left(\sin2x+1\right)-2\sin x.\cos x-2\cos^2x\)
\(=2\cos x\left(2\sin x.\cos x+\sin^2x+\cos^2x\right)-2\cos x\left(\sin x+\cos x\right)\)
\(=2\cos x\left(\sin x+\cos x\right)^2-2\cos x\left(\sin x+\cos x\right)\)
\(=2\cos x\left(\sin x+\cos x\right)\left(\sin x+\cos x-1\right)\)
