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

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

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

prove

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

True

Solution steps

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

Manipulating left side

tan^{4} (t) + tan^{2} (t)

Factor

tan^{2} (t) + tan^{4} (t) : tan^{2} (t) (1 + tan^{2} (t))

tan^{2} (t) + tan^{4} (t)

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

tan^{4} (t) = tan^{2} (t) tan^{2} (t)

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

Factor out common term

tan^{2} (t)

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

1 + sec^{2} (t) - 1 : sec^{2} (t)

1 + sec^{2} (t) - 1

Group like terms

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

1 - 1 = 0

= sec^{2} (t)

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

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

Expand

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

(- 1 + sec^{2} (t)) sec^{2} (t)

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

Apply the distributive law:

a (b + c) = ab + a c

a = sec^{2} (t), b = - 1, c = sec^{2} (t)

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

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot sec^{2} (t) + sec^{2} (t) sec^{2} (t)

Simplify

- 1 \cdot sec^{2} (t) + sec^{2} (t) sec^{2} (t) : - sec^{2} (t) + sec^{4} (t)

- 1 \cdot sec^{2} (t) + sec^{2} (t) sec^{2} (t)

1 \cdot sec^{2} (t) = sec^{2} (t)

1 \cdot sec^{2} (t)

Multiply:

1 \cdot sec^{2} (t) = sec^{2} (t)

= sec^{2} (t)

sec^{2} (t) sec^{2} (t) = sec^{4} (t)

sec^{2} (t) sec^{2} (t)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= sec^{2 + 2} (t)

Add the numbers:

2 + 2 = 4

= sec^{4} (t)

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

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

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

= sec^{4} (t) - sec^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

tan^{2} (x) cos^{2} (x) = sin^{2} (x)

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

tan (A) = \frac{sec ( A )}{csc ( A )}

csc (- x) = - csc (x)

sin (β) + cos (β) cot (β) = csc (β)

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