Is (1+tan^2(s))(1-sin^2(s))=1 ?

The answer to whether (1+tan^2(s))(1-sin^2(s))=1 is True

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prove (1+tan^2(s))(1-sin^2(s))=1

prove

(1 + tan^{2} (s)) (1 - sin^{2} (s)) = 1

True

Solution steps

(1 + tan^{2} (s)) (1 - sin^{2} (s)) = 1

Manipulating left side

(1 + tan^{2} (s)) (1 - sin^{2} (s))

Rewrite using trig identities

(1 + tan^{2} (s)) (1 - sin^{2} (s))

Use the Pythagorean identity:

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

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

= (1 + tan^{2} (s)) cos^{2} (s)

= (1 + tan^{2} (s)) cos^{2} (s)

Express with sin, cos

(1 + tan^{2} (s)) cos^{2} (s)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Join

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

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

Convert element to fraction:

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

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

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

Multiply:

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (s)

= cos^{2} (s) + sin^{2} (s)

= cos^{2} (s) + sin^{2} (s)

= cos^{2} (s) + sin^{2} (s)

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

8 - 4 csc^{2} (z) = 4 - 4 cot^{2} (z)

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

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

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

sin (θ) = sin (θ - 2 π)

Is (1+tan^2(s))(1-sin^2(s))=1 ?
The answer to whether (1+tan^2(s))(1-sin^2(s))=1 is True