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

Rewrite

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

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

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

Apply exponent rule:

a^{b c} = (a^{b})^{c}

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

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

Apply exponent rule:

a^{b c} = (a^{b})^{c}

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

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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)) (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

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

Use the Pythagorean identity:

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

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

Simplify

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

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

Group like terms

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

Add similar elements:

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

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

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

Expand

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

(sec (t) + tan (t)) (sec (t) - tan (t))

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = sec (t), b = tan (t)

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

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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