How does tan^4x-sec^4x equal -2tan^2x-1
If someone could please explain to me then that would be awesome. Thank you ^^
tan^4 - sec^4
= (tan^2-sec^2)(tan^2+sec^2)
= (tan^2-(1+tan^2))(tan^2+(1+tan^2))
= (-1)(1+2tan^2)
= -2tan^2-1
Reiny, imagine the missing x's are present!
Ohhh that makes a lot of sense, thank you!
Steve, as long as they are not imaginary.
To solve this equation, we can start by using the identity: tan^2(x) + 1 = sec^2(x). This identity relates the trigonometric functions tangent (tan) and secant (sec) with the squared values.
Now, let's substitute sec^2(x) with (tan^2(x) + 1) in the equation.
tan^4(x) - sec^4(x) = tan^4(x) - (tan^2(x) + 1)^2
Expanding (tan^2(x) + 1)^2, we have:
tan^4(x) - sec^4(x) = tan^4(x) - (tan^2(x) + 1)(tan^2(x) + 1)
Now, let's simplify further by distributing:
tan^4(x) - sec^4(x) = tan^4(x) - (tan^4(x) + 2tan^2(x) + 1)
Combining like terms, we have:
tan^4(x) - sec^4(x) = tan^4(x) - tan^4(x) - 2tan^2(x) - 1
Canceling out the tan^4(x) terms, we get:
tan^4(x) - sec^4(x) = -2tan^2(x) - 1
Therefore, tan^4(x) - sec^4(x) is equal to -2tan^2(x) - 1.