Is sec^4(θ)-tan^4(θ)=2tan^2(θ)+1 ?

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

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prove sec^4(θ)-tan^4(θ)=2tan^2(θ)+1

prove

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

True

Solution steps

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

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^{4} (θ) - tan^{4} (θ)

Rewrite

sec^{4} (θ) - tan^{4} (θ)

(sec^{2} (θ))^{2} - (tan^{2} (θ))^{2}

sec^{4} (θ) - tan^{4} (θ)

Apply exponent rule:

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

sec^{4} (θ) = (sec^{2} (θ))^{2}

= (sec^{2} (θ))^{2} - tan^{4} (θ)

Apply exponent rule:

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

tan^{4} (θ) = (tan^{2} (θ))^{2}

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

sec^{2} (θ) - tan^{2} (θ)

Apply Difference of Two Squares Formula:

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

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)) (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

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

Use the Pythagorean identity:

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

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

Simplify

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

tan^{2} (θ) + 1 + tan^{2} (θ)

Group like terms

= tan^{2} (θ) + tan^{2} (θ) + 1

Add similar elements:

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

= 2 tan^{2} (θ) + 1

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

Expand

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

(sec (θ) + tan (θ)) (sec (θ) - tan (θ))

Apply Difference of Two Squares Formula:

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

a = sec (θ), b = tan (θ)

= sec^{2} (θ) - tan^{2} (θ)

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

Use the Pythagorean identity:

sec^{2} (θ) = tan^{2} (θ) + 1

sec^{2} (θ) - tan^{2} (θ) = 1

= 1 \cdot (2 tan^{2} (θ) + 1)

Simplify

= 2 tan^{2} (θ) + 1

= 2 tan^{2} (θ) + 1

We showed that the two sides could take the same form

\Rightarrow True

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