Verify the following identity

alexcross · #1 $Old$ March 4th, 2009, 07:50 PM

tan²A+csc²A = cot²A+sec²A

AND

secA-1 = sin²(A)/cosA+cos²A

How would these be done?

red_dog · #2 $Old$ March 5th, 2009, 01:52 AM

1) $\tan^2A+\csc^2A=\frac{\sin^2A}{\cos^2A}+\frac{1}{\sin^2A}=\frac{\sin^4A+\cos^2A}{\sin^2A\cos^2A}=$

$=\frac{(1-\sin^2A)^2+\cos^2A}{\sin^2A\cos^2A}=\frac{1-\cos^2A+\cos^4A}{\sin^2A\cos^2A}=$

$=\frac{\sin^2A+\cos^4A}{\sin^2A\cos^2A}=\frac{1}{\cos^2A}+\frac{\cos^2A}{\sin^2A}=\sec^2A+\cot^2A$

2) $\frac{\sin^2A}{\cos A+\cos^A}=\frac{1-\cos^2A}{\cos A(1+\cos A)}=\frac{(1-\cos A)(1+\cos A)}{\cos A(1+\cos A)}=$

$=\frac{1-\cos A}{\cos A}=\sec A-1$