Is 1+(cos^2(x))/(sin^2(x))=csc^2(x) ?

The answer to whether 1+(cos^2(x))/(sin^2(x))=csc^2(x) is True

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prove 1+(cos^2(x))/(sin^2(x))=csc^2(x)

prove

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

True

Solution steps

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

Manipulating right side

csc^{2} (x)

Rewrite using trig identities

csc^{2} (x)

Use the Pythagorean identity:

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

= 1 + cot^{2} (x)

= 1 + cot^{2} (x)

Express with sin, cos

1 + cot^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Convert element to fraction:

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

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

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

Multiply:

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

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

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

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

Expand

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

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

Apply the fraction rule:

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

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

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

Apply rule

\frac{a}{a} = 1

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

\frac{sin ( 6 0 ^{\circ} )}{cos ( 6 0 ^{\circ} )} = tan (6 0^{\circ})

cos (x) csc (x) = cos (x)

Is 1+(cos^2(x))/(sin^2(x))=csc^2(x) ?
The answer to whether 1+(cos^2(x))/(sin^2(x))=csc^2(x) is True