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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Apply minus-plus rules

- (- a) = a

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

Simplify

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

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

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

1 \cdot sin^{2} (x)

Multiply:

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

= sin^{2} (x)

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

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

Apply exponent rule:

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

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

= sin^{2 + 2} (x)

Add the numbers:

2 + 2 = 4

= sin^{4} (x)

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

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

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

Add similar elements:

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

= sin^{4} (x)

= sin^{4} (x)

= sin^{4} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (2 θ) = \frac{1}{2}

\frac{1}{sin ( x )} - \frac{1}{tan ( x )} = tan (\frac{1}{2} x)

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

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

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

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