Question

Tan^2x+sec^2x=1 for all values of x.
true or false?

Answer 1

The trigonometrical identity says : 1+tan^2x = sec^2 x that means

sec^2 x - tan^2 x = 1

However in our question, its tan^2 x + sec^2 x or say sec^2 x + tan^2 x = 1 which is completely against the identity.

Hense answer is False.

Additional information: There is, with a little difference, as equation which is true ie tan^2 x + sec x = 1 for some values of x.

Answer 2

This is not true.

`\tan^2x+\sec^2x=(\sin^2x)/(\cos^2x)+\frac1{\cos^2x}=(1-\cos^2x+1)/(\cos^2x)=2\sec^2x-1`

`2\sec^2x-1=1\implies\sec^2x=1\implies\sec x=\pm1\implies x=n\pi`

where is
`n` is any integer. So suppose we pick some value of
`x` other than these, say
`x=\frac\pi4`. Then

`\tan^2\frac\pi4+\sec^2\frac\pi4=1+2=3\\eq1`