Mathematics: Prove or disprove that csc^4x − cot^4 x is equal to 2csc^2x − 1

Prove or disprove that csc^4x − cot^4 x is equal

Ответ:

Goldenstate32

13.03.2022

$^{(1)}\csc x=\dfrac{1}{\sin x}\\\\^{(2)}\sin^2x+\cos^2x=1\to ^{(3)}\sin^2x=1-\cos^2x\to ^{(4)}\cos^2x=1-\sin^2x\\\\^{(5)}\cot x=\dfrac{\cos x}{\sin x}\\\\^{(6)} a^2-b^2=(a-b)(a+b)$

$\csc^4x-\cot^4x=2\csc^2x-1\\\\L_s=^{(1)}\dfrac{1}{\sin^4x} -^{(5)}\dfrac{\cos^4x}{\sin^4x}=\dfrac{1-\cos^4x}{\sin^4x}=\dfrac{1^2-(\cos^2x)^2}{\sin^4x}$

$=^{(6)}\dfrac{(1-\cos^2x)(1+\cos^2x)}{\sin^4x}=^{(3)}\dfrac{\sin^2x(1+\cos^2x)}{\sin^4x}$
$=\dfrac{1+\cos^2x}{\sin^2x}=\dfrac{1}{\sin^2x}+\dfrac{\cos^2x}{\sin^2x}=\dfrac{1}{\sin^2x}+^{(4)}\dfrac{1-\sin^2x}{\sin^2x}\\\\=\dfrac{1}{\sin^2x}+\dfrac{1}{\sin^2x}-\dfrac{\sin^2x}{\sin^2x}=\dfrac{2}{\sin^2x}-1=R_s$