Question

Sec^6x(secxtanx)-sec^4(secxtanx)=sec^5xtan^3x
Verify the trigonometric identity

Answer 1

I guess you mean

`\sec^6x(\sec x\tan x)-\sec^4x(\sec x\tan x)=\sec^5x\tan^3x`

On the left side, we have a common factor of
`\sec^4x(\sec x\tan x)=\sec^5x\tan x`, so that

`\sec^6x(\sec x\tan x)-\sec^4x(\sec x\tan x)=\sec^5x\tan x(\sec^2x-1)`

Recall that

`\sec^2x=1+\tan^2x`

from which it follows that

`\sec^5x\tan x(\sec^2x-1)=\sec^5x\tan x\tan^2x=\sec^5x\tan^3x`