Is (1-tan(x))^2=sec^2(x)-2tan(x) ?

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Expand

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

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

Apply Perfect Square Formula:

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

a = 1, b = tan (x)

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

Simplify

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

Multiply the numbers:

2 \cdot 1 = 2

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

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

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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