Is sec^4(θ)-tan^4(θ)=1+2tan^2(θ) ?

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

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prove sec^4(θ)-tan^4(θ)=1+2tan^2(θ)

prove

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

True

Solution steps

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

Manipulating left side

sec^{4} (θ) - tan^{4} (θ)

Factor

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

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

Rewrite using trig identities

= 2 tan^{2} (θ) + 1

= 1 + 2 tan^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

sin (0) tan (0) + cos (0) = sec^{2} (0)

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

sin (3 x) = (sin (x)) (3 - 4 sin^{2} (x))

sec (a) + tan (a) = \frac{1}{sec ( a )} + tan (a)

cos (3 a) = 4 cos^{3} (a) - 3 cos (a)

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