Is tan^2(u)-sin^2(u)=tan^2(u)sin^2(u) ?

The answer to whether tan^2(u)-sin^2(u)=tan^2(u)sin^2(u) is True

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prove tan^2(u)-sin^2(u)=tan^2(u)sin^2(u)

prove

tan^{2} (u) - sin^{2} (u) = tan^{2} (u) sin^{2} (u)

True

Solution steps

tan^{2} (u) - sin^{2} (u) = tan^{2} (u) sin^{2} (u)

Manipulating right side

tan^{2} (u) sin^{2} (u)

Rewrite using trig identities

tan^{2} (u) sin^{2} (u)

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= tan^{2} (u) (1 - cos^{2} (u))

= tan^{2} (u) (1 - cos^{2} (u))

Expand

(1 - cos^{2} (u)) tan^{2} (u) : tan^{2} (u) - tan^{2} (u) cos^{2} (u)

(1 - cos^{2} (u)) tan^{2} (u)

= tan^{2} (u) (1 - cos^{2} (u))

Apply the distributive law:

a (b - c) = ab - a c

a = tan^{2} (u), b = 1, c = cos^{2} (u)

= tan^{2} (u) \cdot 1 - tan^{2} (u) cos^{2} (u)

= 1 \cdot tan^{2} (u) - tan^{2} (u) cos^{2} (u)

Multiply:

1 \cdot tan^{2} (u) = tan^{2} (u)

= tan^{2} (u) - tan^{2} (u) cos^{2} (u)

= tan^{2} (u) - cos^{2} (u) tan^{2} (u)

Rewrite using trig identities

tan^{2} (u) - cos^{2} (u) tan^{2} (u)

Use the basic trigonometric identity:

tan (u) = \frac{sin ( u )}{cos ( u )}

= tan^{2} (u) - cos^{2} (u) (\frac{sin ( u )}{cos ( u )})^{2}

Simplify

cos^{2} (u) (\frac{sin ( u )}{cos ( u )})^{2} : sin^{2} (u)

cos^{2} (u) (\frac{sin ( u )}{cos ( u )})^{2}

(\frac{sin ( u )}{cos ( u )})^{2} = \frac{sin ^{2} ( u )}{cos ^{2} ( u )}

(\frac{sin ( u )}{cos ( u )})^{2}

Apply exponent rule:

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

= \frac{sin ^{2} ( u )}{cos ^{2} ( u )}

= \frac{sin ^{2} ( u )}{cos ^{2} ( u )} cos^{2} (u)

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{sin ^{2} ( u ) cos ^{2} ( u )}{cos ^{2} ( u )}

Cancel the common factor:

cos^{2} (u)

= sin^{2} (u)

= tan^{2} (u) - sin^{2} (u)

= tan^{2} (u) - sin^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Is tan^2(u)-sin^2(u)=tan^2(u)sin^2(u) ?
The answer to whether tan^2(u)-sin^2(u)=tan^2(u)sin^2(u) is True