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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Expand

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

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Multiply:

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (x)

= sin^{2} (x)

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{sin ( x )}{1 - cos ( x )} - cot (x) = csc (x)

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

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

sin (2 x) = 2 sin (x) cos (x)

\frac{csc ^{2} ( x )}{cot ( x )} = csc (x) sec (x)

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