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

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

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

prove

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

True

Solution steps

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

(sin (2 θ)) (tan (θ)) + cos (2 θ) = 1

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

cot^{2} (θ) = \frac{csc ^{2} ( θ )}{sec ^{2} ( θ )}

\frac{1}{sec ( θ )} = arcsec (θ)

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