Question

evaluate (cosec theta-cot theta)^2 = 1-cos theta/1+costheta​

Answer 1

$\mathfrak{\huge{\underline{\pink{ANSWER :-}}}}$

$=> (cosec x-cot x)^2 = ((1-cos x)/(1+cos x))\\ \\ \\\implies [(1/sin x) - (cos x/sin x)]^2 = [2.(sin (x/2))^2]/[2.(cos (x/2))^2]\\ \\ \\\implies ((1-cos x)/sin x)^2 = [tan(x/2)]^2\\ \\ \\\implies {[2.(sin (x/2))^2]/[2.sin(x/2).cos (x/2)]}^2 = [tan(x/2)]^2\\ \\ \\\implies [sin(x/2) / cos(x/2) ]^2 = [tan(x/2)]^2\\ \\ \\ \implies [tan(x/2)]^2 = [tan(x/2)]^2$

Here , 1 = 1

Hence proved

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Answer 2

$\mathfrak{\huge{\underline{\pink{ANSWER :-}}}}$

$=> (cosec x-cot x)^2 = ((1-cos x)/(1+cos x))\\ \\ \\\implies [(1/sin x) - (cos x/sin x)]^2 = [2.(sin (x/2))^2]/[2.(cos (x/2))^2]\\ \\ \\\implies ((1-cos x)/sin x)^2 = [tan(x/2)]^2\\ \\ \\\implies {[2.(sin (x/2))^2]/[2.sin(x/2).cos (x/2)]}^2 = [tan(x/2)]^2\\ \\ \\\implies [sin(x/2) / cos(x/2) ]^2 = [tan(x/2)]^2\\ \\ \\ \implies [tan(x/2)]^2 = [tan(x/2)]^2$

Here , 1 = 1

Hence proved

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