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

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

prove

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^{4} (x) + tan^{4} (x)

Rewrite

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

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

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

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

Apply Difference of Two Squares Formula:

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

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

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

Factor

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

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

Apply Difference of Two Squares Formula:

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

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

= (tan (x) + sec (x)) (tan (x) - sec (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

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

Use the Pythagorean identity:

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

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

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

Add similar elements:

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

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

Expand

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

(tan (x) + sec (x)) (tan (x) - sec (x))

Apply Difference of Two Squares Formula:

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

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

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

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Remove parentheses:

(- a) = - a

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

Multiply:

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

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

Distribute parentheses

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

Apply minus-plus rules

- (- a) = a, - (a) = - a

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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