Is cot^2(a)+1= 1/(sin^2(a)) ?

The answer to whether cot^2(a)+1= 1/(sin^2(a)) is True

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prove cot^2(a)+1= 1/(sin^2(a))

prove

cot^{2} (a) + 1 = \frac{1}{sin ^{2} ( a )}

True

Solution steps

cot^{2} (a) + 1 = \frac{1}{sin ^{2} ( a )}

Manipulating left side

cot^{2} (a) + 1

Express with sin, cos

1 + cot^{2} (a)

Use the basic trigonometric identity:

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

= 1 + (\frac{cos ( a )}{sin ( a )})^{2}

Simplify

1 + (\frac{cos ( a )}{sin ( a )})^{2} : \frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

1 + (\frac{cos ( a )}{sin ( a )})^{2}

Apply exponent rule:

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

= 1 + \frac{cos ^{2} ( a )}{sin ^{2} ( a )}

Convert element to fraction:

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

= \frac{1 \cdot sin ^{2} ( a )}{sin ^{2} ( a )} + \frac{cos ^{2} ( a )}{sin ^{2} ( a )}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

Multiply:

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

= \frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

= \frac{sin ^{2} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

= \frac{cos ^{2} ( a ) + sin ^{2} ( a )}{sin ^{2} ( a )}

Rewrite using trig identities

\frac{cos ^{2} ( a ) + sin ^{2} ( a )}{sin ^{2} ( a )}

Use the Pythagorean identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sin (θ) sec (θ) cot (θ) = 1

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

3 sin^{2} (x) + 4 cos^{2} (x) = 3 + cos^{2} (x)

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

sec (θ) + tan (θ) = \frac{cos ( θ )}{1 - sin ( θ )}

Is cot^2(a)+1= 1/(sin^2(a)) ?
The answer to whether cot^2(a)+1= 1/(sin^2(a)) is True