Is tan^4(x)-sec^4(x)=-2tan^2(x)-1 ?

The answer to whether tan^4(x)-sec^4(x)=-2tan^2(x)-1 is True

Popular >

prove tan^4(x)-sec^4(x)=-2tan^2(x)-1

prove

tan^{4} (x) - sec^{4} (x) = - 2 tan^{2} (x) - 1

True

Solution steps

tan^{4} (x) - sec^{4} (x) = - 2 tan^{2} (x) - 1

Manipulating left side

tan^{4} (x) - sec^{4} (x)

Factor

- sec^{4} (x) + tan^{4} (x) : (tan^{2} (x) + sec^{2} (x)) (tan (x) + sec (x)) (tan (x) - sec (x))

= (- sec (x) + tan (x)) (sec (x) + tan (x)) (sec^{2} (x) + tan^{2} (x))

Rewrite using trig identities

= - 2 tan^{2} (x) - 1

We showed that the two sides could take the same form

\Rightarrow True

\frac{1}{1 + cos ( x )} = csc^{2} (x) - csc (x) cot (x)

sin^{2} (2 x) = 1 - cos^{2} (2 x)

sin (\frac{A}{2}) = \frac{1}{2} sin (A)

\frac{cos ( θ )}{1 - sin ( θ )} = tan (θ) + sec (θ)

\frac{2 cos ( 2 θ )}{sin ( 2 θ )} = cot (θ) - tan (θ)

Is tan^4(x)-sec^4(x)=-2tan^2(x)-1 ?
The answer to whether tan^4(x)-sec^4(x)=-2tan^2(x)-1 is True