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

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

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

prove

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

True

Solution steps

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

csc (x) + 1 = \frac{cot ( x )}{sec ( x ) - tan ( x )}

sin (x) sec^{2} (x) = \frac{tan ( x )}{cos ( x )}

\frac{1 + sec ^{2} ( x )}{sec ^{2} ( x )} = 1 + cos^{2} (x)

sin (2 θ) = 2 cot (θ) sin^{2} (θ)

tan (θ) + 1 = sec (θ)

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