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

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

Popular >

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

prove

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

True

Solution steps

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

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

8 csc^{2} (x) - 3 cot^{2} (x) = 3 + 5 csc^{2} (x)

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