The equations are solved below using trigonometric identities.
What are trigonometric identities?
Trigonometric identities are equalities that involve trigonometric functions and are true for every value of the occurring variables for which both sides of the equality are defined.
[tex]\frac{sin(x)}{1-cos(x)} -cot(x) = cosec(x) \\\\LHS= \\\\=\frac{sin(x)}{1-cos(x)} -cot(x) \\\\=\frac{sin(x)}{1-cos(x)} -\frac{cos(x)}{sin(x)}\\ \\\frac{sin^{2}(x) - (1-cos(x))cos(x) }{sin(x)(1-cos(x))}\\\\=\frac{sin^{2}(x)-cos(x) +cos^{2}(x) }{sin(x)(1-cos(x))}\\\\=\frac{1-cos(x)}{sin(x)(1-cos(x))}\\ \\=\frac{1}{sin(x)} \\\\=cosec(x)\\\\RHS=\\\\=cosec(x)[/tex]
[tex]cosec(x)^{2} - 2cosec(x)cot(x) + cot(x)^{2} = tan^{2}(\frac{x}{2})\\\\LHS= \\\\= cosec(x)^{2} - 2cosec(x)cot(x) + cot(x)^{2} \\\\=( cosec(x) - cot(x) )^{2} \\\\=(\frac{1-cos(x)}{sin(x)}) ^{2}\\\\=(\frac{(1-cos(x))(1+cos(x))}{sin(x)(1+cos(x))}) ^{2}\\\\=(\frac{sin(x)^{2} }{sin(x)(1+cos(x))}) ^{2}\\\\=(\frac{sin(x) }{1+cos(x)}) ^{2}\\\\=(\frac{2sin(x/2)cos(x/2) }{1+2cos^{2}(x/2)-1}) ^{2}\\\\= (\frac{sin(x/2)cos(x/2) }{cos^{2}(x/2)}) ^{2}\\\\= tan^{2}\frac{x}{2}\\\\RHS = \\\\=tan^{2}\frac{x}{2}[/tex]
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