Mathematics: Simplify the trigonomic expression 1/1+ sin

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Simplify the trigonomic expression 1/1+ sin

trinidymwilga

10.01.2022

$\sec^2\theta-\tan\theta\sec\theta$

Step-by-step explanation:

We have to simplify the trigonometric expression $\frac{1}{1+\sin\theta}$

Let us multiply the numerator and denominator by $1-\sin\theta$

$\frac{1}{1+\sin\theta}\times\frac{1-\sin\theta}{1-\sin\theta}$

In the denominator apply the formula for difference of squares

a^2-b^2=(a+b)(a-b)

Thus, the denominator will become

$(1+\sin\theta)(1-\sin\theta)=1-\sin^2\theta$

Thus, the expression is

$\frac{1-\sin\theta}{1-\sin^2\theta}$

Using the relation $1-\sin^2\theta=\cos^2\theta$

$\frac{1-\sin\theta}{\cos^2\theta}$

We can rewrite this expression as

$\frac{1}{\cos^2\theta}-\frac{\sin\theta}{\cos^2\theta}$

Finally, the simplified expression is

$\sec^2\theta-\tan\theta\sec\theta$

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