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

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

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prove sec^4(t)-tan^4(t)=1+2tan^2(t)

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite using trig identities

= 2 tan^{2} (t) + 1

= 1 + 2 tan^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

cos (4 θ) = cos^{2} (2 θ) - sin^{2} (2 θ)

sec (x) csc (x) - cot (x) = tan (x)

tan (θ) sin (2 θ) = 2 - 2 cos^{2} (θ)

- 30 cos (3 x) sin (3 x) = - 15 sin (6 x)

2 sin^{2} (θ) - 1 = sin^{4} (θ) - cos^{4} (θ)

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