Question

Precalculus please help me right answers only

Answer 1

Consider all options.

A.

`\sec^2x-\csc^2x=1.`

Note that

`\sec x=(1)/(\cos x),\\ \\\csc x=(1)/(\sin x).`

Then

`\sec^2x-\csc^2x=(1)/(\cos^2x)-(1)/(\sin^2x)=(\sin^2x-\cos^2x)/(\sin^2x\cos^2x)=(-4\cos (2x))/(\sin^2(2x))=\\ \\=-4(\cot (2x))/(\sin (2x))\\eq 1.`

This option is false.

B.

Since
`\cos (2x)=\cos^2x-\sin^2x,` then
`\sin^2x-\cos^2 x=-\cos (2x)\\eq 1.`

This option is false.

C.

Consider

`1+\cot^2x=1+(\cos^2x)/(\sin^2x)=(\sin^2x+\cos^2x)/(\sin^2x)=(1)/(\sin^2x)=\csc^2x.`

This option is true.

D.

`\sec^2x-\tan^2x=(1)/(\cos^2x)-(\sin^2x)/(\cos^2x)=(1-\sin^2x)/(\cos^2x)=(\cos^2x)/(\cos^2x)=1.`

This option is true.

Answer: False A and B.