Answer:
[tex]\sec^2\theta-\tan\theta\sec\theta[/tex]
Step-by-step explanation:
We have to simplify the trigonometric expression [tex]\frac{1}{1+\sin\theta}[/tex]
Let us multiply the numerator and denominator by [tex]1-\sin\theta[/tex]
[tex]\frac{1}{1+\sin\theta}\times\frac{1-\sin\theta}{1-\sin\theta}[/tex]
In the denominator apply the formula for difference of squares
[tex]a^2-b^2=(a+b)(a-b)[/tex]
Thus, the denominator will become
[tex](1+\sin\theta)(1-\sin\theta)=1-\sin^2\theta[/tex]
Thus, the expression is
[tex]\frac{1-\sin\theta}{1-\sin^2\theta}[/tex]
Using the relation [tex]1-\sin^2\theta=\cos^2\theta[/tex]
[tex]\frac{1-\sin\theta}{\cos^2\theta}[/tex]
We can rewrite this expression as
[tex]\frac{1}{\cos^2\theta}-\frac{\sin\theta}{\cos^2\theta}[/tex]
Finally, the simplified expression is
[tex]\sec^2\theta-\tan\theta\sec\theta[/tex]
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.
