Is cos^7(x)=(1-sin^2(x))^3cos(x) ?

The answer to whether cos^7(x)=(1-sin^2(x))^3cos(x) is True

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prove cos^7(x)=(1-sin^2(x))^3cos(x)

prove

cos^{7} (x) = (1 - sin^{2} (x))^{3} cos (x)

True

Solution steps

cos^{7} (x) = (1 - sin^{2} (x))^{3} cos (x)

Manipulating right side

(1 - sin^{2} (x))^{3} cos (x)

Rewrite using trig identities

(1 - sin^{2} (x))^{3} cos (x)

Use the Pythagorean identity:

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

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

= (cos^{2} (x))^{3} cos (x)

Simplify

(cos^{2} (x))^{3} cos (x) : cos^{7} (x)

(cos^{2} (x))^{3} cos (x)

(cos^{2} (x))^{3} = cos^{6} (x)

(cos^{2} (x))^{3}

Apply exponent rule:

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

= cos^{2 \cdot 3} (x)

Multiply the numbers:

2 \cdot 3 = 6

= cos^{6} (x)

= cos^{6} (x) cos (x)

Apply exponent rule:

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

cos^{6} (x) cos (x) = cos^{6 + 1} (x)

= cos^{6 + 1} (x)

Add the numbers:

6 + 1 = 7

= cos^{7} (x)

= cos^{7} (x)

= cos^{7} (x)

We showed that the two sides could take the same form

\Rightarrow True

(1 + tan (θ))^{2} = sec^{2} (θ) + 2 tan (θ)

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

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

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

sin^{2} (θ) cot^{2} (θ) = 1

Is cos^7(x)=(1-sin^2(x))^3cos(x) ?
The answer to whether cos^7(x)=(1-sin^2(x))^3cos(x) is True