Question

Simplify the trigonomic expression 1/1+ sin

Answer 1

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

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.

Answer 2

`\sec^2\theta-\tan\theta\sec\theta`

Explanation:

We have to simplify the trigonometric expression
`(1)/(1+\sin\theta)`

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

`(1)/(1+\sin\theta)*(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

`(1-\sin\theta)/(1-\sin^2\theta)`

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

`(1-\sin\theta)/(\cos^2\theta)`

We can rewrite this expression as

`(1)/(\cos^2\theta)-(\sin\theta)/(\cos^2\theta)`

Finally, the simplified expression is

`\sec^2\theta-\tan\theta\sec\theta`