Is sin^2(A)-sin^2(A)cos^2(A)=sin^4(A) ?

The answer to whether sin^2(A)-sin^2(A)cos^2(A)=sin^4(A) is True

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prove sin^2(A)-sin^2(A)cos^2(A)=sin^4(A)

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

= sin^{2} (A) - sin^{2} (A) (1 - sin^{2} (A))

Simplify

sin^{2} (A) - sin^{2} (A) (1 - sin^{2} (A)) : sin^{4} (A)

sin^{2} (A) - sin^{2} (A) (1 - sin^{2} (A))

Expand

- sin^{2} (A) (1 - sin^{2} (A)) : - sin^{2} (A) + sin^{4} (A)

- sin^{2} (A) (1 - sin^{2} (A))

Apply the distributive law:

a (b - c) = ab - a c

a = - sin^{2} (A), b = 1, c = sin^{2} (A)

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

Apply minus-plus rules

- (- a) = a

= - 1 \cdot sin^{2} (A) + sin^{2} (A) sin^{2} (A)

Simplify

- 1 \cdot sin^{2} (A) + sin^{2} (A) sin^{2} (A) : - sin^{2} (A) + sin^{4} (A)

- 1 \cdot sin^{2} (A) + sin^{2} (A) sin^{2} (A)

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

1 \cdot sin^{2} (A)

Multiply:

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

= sin^{2} (A)

sin^{2} (A) sin^{2} (A) = sin^{4} (A)

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

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

sin^{2} (A) sin^{2} (A) = sin^{2 + 2} (A)

= sin^{2 + 2} (A)

Add the numbers:

2 + 2 = 4

= sin^{4} (A)

= - sin^{2} (A) + sin^{4} (A)

= - sin^{2} (A) + sin^{4} (A)

= sin^{2} (A) - sin^{2} (A) + sin^{4} (A)

Add similar elements:

sin^{2} (A) - sin^{2} (A) = 0

= sin^{4} (A)

= sin^{4} (A)

= sin^{4} (A)

We showed that the two sides could take the same form

\Rightarrow True

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

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

sin (a) sec (a) tan (a) + 1 = sec^{2} (a)

\frac{tan ( y ) + cot ( y )}{csc ( y )} = sec (y)

\frac{sec ^{2} ( a ) - 1}{sec ^{2} ( a )} = sin^{2} (a)

Is sin^2(A)-sin^2(A)cos^2(A)=sin^4(A) ?
The answer to whether sin^2(A)-sin^2(A)cos^2(A)=sin^4(A) is True