Is 1/(1+sin(t))=(1-sin(t))/(cos^2(t)) ?

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

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prove 1/(1+sin(t))=(1-sin(t))/(cos^2(t))

prove

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

True

Solution steps

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

Manipulating right side

\frac{1 - sin ( t )}{cos ^{2} ( t )}

Rewrite using trig identities

\frac{1 - sin ( t )}{cos ^{2} ( t )}

Use the Pythagorean identity:

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

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

= \frac{1 - sin ( t )}{1 - sin ^{2} ( t )}

Simplify

\frac{1 - sin ( t )}{1 - sin ^{2} ( t )} : \frac{1}{sin ( t ) + 1}

\frac{1 - sin ( t )}{1 - sin ^{2} ( t )}

Factor

1 - sin^{2} (t) : - (sin (t) + 1) (sin (t) - 1)

1 - sin^{2} (t)

Factor out common term

- 1

= - (sin^{2} (t) - 1)

Factor

sin^{2} (t) - 1 : (sin (t) + 1) (sin (t) - 1)

sin^{2} (t) - 1

Rewrite

1

1^{2}

= sin^{2} (t) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

sin^{2} (t) - 1^{2} = (sin (t) + 1) (sin (t) - 1)

= (sin (t) + 1) (sin (t) - 1)

= - (sin (t) + 1) (sin (t) - 1)

= - \frac{1 - sin ( t )}{( sin ( t ) + 1 ) ( sin ( t ) - 1 )}

Cancel

- \frac{1 - sin ( t )}{( sin ( t ) + 1 ) ( sin ( t ) - 1 )} : \frac{1}{sin ( t ) + 1}

- \frac{1 - sin ( t )}{( sin ( t ) + 1 ) ( sin ( t ) - 1 )}

sin (t) - 1 = - (- sin (t) + 1)

= - \frac{- sin ( t ) + 1}{- ( sin ( t ) + 1 ) ( - sin ( t ) + 1 )}

Refine

= \frac{1 - sin ( t )}{( sin ( t ) + 1 ) ( - sin ( t ) + 1 )}

Cancel the common factor:

1 - sin (t)

= \frac{1}{sin ( t ) + 1}

= \frac{1}{sin ( t ) + 1}

= \frac{1}{sin ( t ) + 1}

= \frac{1}{sin ( t ) + 1}

= \frac{1}{1 + sin ( t )}

We showed that the two sides could take the same form

\Rightarrow True

1 + tan^{2} (A) = sec^{3} (A) cos (A)

sec (0) - cos (0) = tan (0) sin (0)

sin (x) - csc (x) = - cot (x) cos (x)

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

tan (x) \cdot cos (x) \cdot csc (x) = 1

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