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

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

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

prove

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

True

Solution steps

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

Expand

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

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

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

Multiply:

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (A)

= sin^{2} (A)

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

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

We showed that the two sides could take the same form

\Rightarrow True

cos (θ) sec (\frac{π}{2} - θ) = cot (θ)

4 cos^{4} (x) - 4 sin^{4} (x) = 4 cos (2 x)

csc (x) (1 + sin (x)) = 1 + csc (x)

sin (x + y) sin (x - y) = sin (2 x) - sin (2 y)

csc^{2} (x) = 1 - cot^{2} (x)

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