Is (1-tan(θ))^2=sec^2(θ)-2tan(θ) ?

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

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prove (1-tan(θ))^2=sec^2(θ)-2tan(θ)

prove

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

True

Solution steps

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

Manipulating left side

(1 - tan (θ))^{2}

Expand

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

(1 - tan (θ))^{2}

Apply Perfect Square Formula:

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

a = 1, b = tan (θ)

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

Simplify

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

1^{2} - 2 \cdot 1 \cdot tan (θ) + tan^{2} (θ)

Apply rule

1^{a} = 1

1^{2} = 1

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

Multiply the numbers:

2 \cdot 1 = 2

= 1 - 2 tan (θ) + tan^{2} (θ)

= 1 - 2 tan (θ) + tan^{2} (θ)

= 1 + tan^{2} (θ) - 2 tan (θ)

Rewrite using trig identities

1 + tan^{2} (θ) - 2 tan (θ)

Use the Pythagorean identity:

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

= - 2 tan (θ) + sec^{2} (θ)

= - 2 tan (θ) + sec^{2} (θ)

= sec^{2} (θ) - 2 tan (θ)

We showed that the two sides could take the same form

\Rightarrow True

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