Is 1-tan^4(θ)=2sec^2(θ)-sec^4(θ) ?

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

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prove 1-tan^4(θ)=2sec^2(θ)-sec^4(θ)

prove

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

True

Solution steps

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

Manipulating right side

2 sec^{2} (θ) - sec^{4} (θ)

Factor

- sec^{4} (θ) + 2 sec^{2} (θ) : sec^{2} (θ) (- sec^{2} (θ) + 2)

- sec^{4} (θ) + 2 sec^{2} (θ)

Apply exponent rule:

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

sec^{4} (θ) = sec^{2} (θ) sec^{2} (θ)

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

Factor out common term

sec^{2} (θ)

= sec^{2} (θ) (- sec^{2} (θ) + 2)

= (2 - sec^{2} (θ)) sec^{2} (θ)

Rewrite using trig identities

(2 - sec^{2} (θ)) sec^{2} (θ)

Use the Pythagorean identity:

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

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

Expand

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

2 - (tan^{2} (θ) + 1)

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

- (tan^{2} (θ) + 1)

Distribute parentheses

= - (tan^{2} (θ)) - (1)

Apply minus-plus rules

+ (- a) = - a

= - tan^{2} (θ) - 1

= 2 - tan^{2} (θ) - 1

Simplify

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

2 - tan^{2} (θ) - 1

Group like terms

= - tan^{2} (θ) + 2 - 1

Add/Subtract the numbers:

2 - 1 = 1

= - tan^{2} (θ) + 1

= - tan^{2} (θ) + 1

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

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

Expand

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

(1 + tan^{2} (θ)) (1 - tan^{2} (θ))

Apply Difference of Two Squares Formula:

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

a = 1, b = tan^{2} (θ)

= 1^{2} - (tan^{2} (θ))^{2}

Simplify

1^{2} - (tan^{2} (θ))^{2} : 1 - tan^{4} (θ)

1^{2} - (tan^{2} (θ))^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - (tan^{2} (θ))^{2}

(tan^{2} (θ))^{2} = tan^{4} (θ)

(tan^{2} (θ))^{2}

Apply exponent rule:

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

= tan^{2 \cdot 2} (θ)

Multiply the numbers:

2 \cdot 2 = 4

= tan^{4} (θ)

= 1 - tan^{4} (θ)

= 1 - tan^{4} (θ)

= 1 - tan^{4} (θ)

We showed that the two sides could take the same form

\Rightarrow True

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