Is cos(t)cot(t)=(1-sin^2(t))/(sin(t)) ?

The answer to whether cos(t)cot(t)=(1-sin^2(t))/(sin(t)) is True

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prove cos(t)cot(t)=(1-sin^2(t))/(sin(t))

prove

cos (t) cot (t) = \frac{1 - sin ^{2} ( t )}{sin ( t )}

True

Solution steps

cos (t) cot (t) = \frac{1 - sin ^{2} ( t )}{sin ( t )}

Manipulating left side

cos (t) cot (t)

Express with sin, cos

cos (t) cot (t)

Use the basic trigonometric identity:

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

= cos (t) \frac{cos ( t )}{sin ( t )}

Simplify

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

cos (t) \frac{cos ( t )}{sin ( t )}

Multiply fractions:

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

= \frac{cos ( t ) cos ( t )}{sin ( t )}

cos (t) cos (t) = cos^{2} (t)

cos (t) cos (t)

Apply exponent rule:

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

cos (t) cos (t) = cos^{1 + 1} (t)

= cos^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= cos^{2} (t)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

(1 - cos (θ)) (1 + cos (θ)) (1 + cot^{2} (θ)) = 1

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

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

tan (u) cot (u) - cos^{2} (u) = sin^{2} (u)

Is cos(t)cot(t)=(1-sin^2(t))/(sin(t)) ?
The answer to whether cos(t)cot(t)=(1-sin^2(t))/(sin(t)) is True