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MariettaO [177]
3 years ago
6

Simplify the trigonomic expression 1/1+ sin

Mathematics
2 answers:
Softa [21]3 years ago
6 0

Answer:

\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

34kurt3 years ago
6 0

Answer:

sec²θ - tanθ(secθ) is the simplification of given trigonometric expression.

Step-by-step explanation:

We have given the following  trigonometric expression:

1/1+sinθ

We have to simplify this   trigonometric expression.

For this, we multiply and divide the given  trigonometric expression by

1-sinθ.

(1/1+sinθ ) ÷ ( 1-sinθ)/(1-sinθ)

1(1-sinθ) / (1-sinθ)(1+sinθ)

(1-sinθ) / (1-sin²θ)

We know that 1-sin²θ = cos²θ.

We replace the denominator of above trigonometric expression by  cos²θ.

( 1-sinθ) / cos²θ

(1/cos²θ) - (sinθ/cos²θ)

(1/cos²θ) - ( sinθ/cosθ)(1/cosθ)

As we know that 1/cos²θ = sec²θ, sinθ/cosθ = tanθ and 1/cosθ = secθ.So we have,

sec²θ - tanθ(secθ) is the simplification of given trigonometric expression.

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