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Question Number 219868 by Nicholas666 last updated on 02/May/25
Prove that; (d/dx) (((sin^( 2) x)/(1+cot x)) + ((cos^( 2) x)/(1+tan x))) = −cos 2x
$$ \\ $$$$\:\:\:\:\:\:\:\:\:\mathrm{Prove}\:\mathrm{that}; \\ $$$$\:\:\:\:\frac{{d}}{{dx}}\:\left(\frac{\mathrm{sin}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{cot}\:{x}}\:+\:\frac{\mathrm{cos}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{tan}\:{x}}\right)\:=\:−\mathrm{cos}\:\mathrm{2}{x}\:\:\:\: \\ $$$$ \\ $$
Answered by MrGaster last updated on 03/May/25
((sin^( 2) x)/(1+cot x)) + ((cos^( 2) x)/(1+tan x)) ((sin^( 2) x)/(1+cot x))=((sin^2 x)/((sin x+cos x)/(sinx)))=((sin^3 x)/(sin x+cos x)) ((cos^2 x)/(1+tan x))=((cos^2 x)/((cos x+sin x)/(cos x)))=((cos^3 x)/(sin x+cos x)) ((sin^3 x+cos^3 x)/(sin x+cos x))=(((sin x+cos x)(sin^2 x−sin x cos x+cos^2 x))/(sin x+cos x)) sin^2 x−sin x cos x+cos^2 x=1−sin x cos x (d/dx) (((sin^( 2) x)/(1+cot x)) + ((cos^( 2) x)/(1+tan x))) = −cos 2x
$$\frac{\mathrm{sin}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{cot}\:{x}}\:+\:\frac{\mathrm{cos}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{tan}\:{x}} \\ $$$$\frac{\mathrm{sin}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{cot}\:{x}}=\frac{\mathrm{sin}^{\mathrm{2}} {x}}{\frac{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}{\mathrm{sin}{x}}}=\frac{\mathrm{sin}^{\mathrm{3}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}} \\ $$$$\frac{\mathrm{cos}^{\mathrm{2}} {x}}{\mathrm{1}+\mathrm{tan}\:{x}}=\frac{\mathrm{cos}^{\mathrm{2}} {x}}{\frac{\mathrm{cos}\:{x}+\mathrm{sin}\:{x}}{\mathrm{cos}\:{x}}}=\frac{\mathrm{cos}^{\mathrm{3}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}} \\ $$$$\frac{\mathrm{sin}^{\mathrm{3}} {x}+\mathrm{cos}^{\mathrm{3}} {x}}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}}=\frac{\left(\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)\left(\mathrm{sin}^{\mathrm{2}} {x}−\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}+\mathrm{cos}^{\mathrm{2}} {x}\right)}{\mathrm{sin}\:{x}+\mathrm{cos}\:{x}} \\ $$$$\mathrm{sin}^{\mathrm{2}} {x}−\mathrm{sin}\:{x}\:\mathrm{cos}\:{x}+\mathrm{cos}^{\mathrm{2}} {x}=\mathrm{1}−\mathrm{sin}\:{x}\:\mathrm{cos}\:{x} \\ $$$$\:\:\frac{{d}}{{dx}}\:\left(\frac{\mathrm{sin}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{cot}\:{x}}\:+\:\frac{\mathrm{cos}^{\:\mathrm{2}} {x}}{\mathrm{1}+\mathrm{tan}\:{x}}\right)\:=\:−\mathrm{cos}\:\mathrm{2}{x} \\ $$
Commented by Nicholas666 last updated on 03/May/25
thanks
$${thanks} \\ $$

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