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

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

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

prove

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

True

Solution steps

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

Manipulating left side

(tan (x) + 1) (tan (x) - 1)

Expand

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

(- 1 + tan (x)) (1 + tan (x))

Apply Difference of Two Squares Formula:

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

a = tan (x), b = 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= tan^{2} (x) - 1

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

Manipulating right side

sec^{2} (x) - 2

Rewrite using trig identities

sec^{2} (x) - 2

Use the Pythagorean identity:

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

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

Simplify

= tan^{2} (x) - 1

= tan^{2} (x) - 1

We showed that the two sides could take the same form

\Rightarrow True

\frac{1}{csc ( x ) + 1} = \frac{1}{sin ( x ) + 1}

tan (θ) + 1 = \frac{sin ( θ )}{cos ( θ )} + 1

(1 + tan^{2} (α)) cos^{2} (α) = 1

\frac{1}{sin ^{2} ( x )} = \frac{1}{csc ^{2} ( x )}

sin^{2} (θ) = (1 - cos^{2} (θ))

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