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

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

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

prove

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

True

Solution steps

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

Manipulating right side

sec^{2} (x)

Rewrite using trig identities

sec^{2} (x)

Use the Pythagorean identity:

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

= tan^{2} (x) + 1

= tan^{2} (x) + 1

Express with sin, cos

1 + tan^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Convert element to fraction:

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

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

Multiply:

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

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

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

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

Expand

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

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

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

Apply rule

\frac{a}{a} = 1

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

csc (\frac{π}{12}) = sec (\frac{5 π}{12})

1 - \frac{cot ^{2} ( θ )}{csc ^{2} ( θ )} = sin^{2} (θ)

1 = sec^{2} (x)

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