simplify 1/( 1- sin 46) raise to power 2
To simplify the expression (1 / (1 - sin 46))², we can start by simplifying the denominator.
The identity 1 - sin²θ = cos²θ can be applied here.
So, 1 - sin 46 can be written as cos² 46.
Therefore, the expression becomes (1 / cos² 46)².
Now, we can use the property of exponents to simplify further.
(a / b)² = a² / b².
Applying this property to our expression, we have 1² / (cos² 46)².
Simplifying even further, we get 1 / cos⁴ 46.
To simplify the expression (1/(1 - sin 46))^2, we'll start by rewriting it as (1 - sin 46)^(-2) since raising a fraction to the power of -1 is the same as flipping the fraction.
Next, we'll use the property that (a - b)^n is equal to a^n - n * a^(n-1) * b + ... + (-1)^n * b^n.
Applying this property, we get:
(1 - sin 46)^(-2) = 1^(-2) - 2 * 1^(-1) * sin 46 + sin^2(46)
= 1 - 2 * sin 46 + sin^2(46)
Therefore, the expression (1/(1 - sin 46))^2 simplifies to 1 - 2 * sin 46 + sin^2(46).
To simplify the expression (1/(1 - sin 46))^2, we'll break it down into steps:
Step 1: Evaluate the expression inside the parentheses.
In this case, we need to find the value of 1 - sin 46.
Step 2: Simplify the expression inside the parentheses.
To simplify 1 - sin 46, you'll need to use a trigonometric identity. For reference, the identity is: sin(90 - θ) = cos θ.
In this case, since sin(90 - 46) = sin 44 = sin(90 - 46), we can conclude that sin 46 = cos 44.
So, the expression simplifies to 1 - cos 44.
Step 3: Substitute the simplified expression back into the original expression.
Now that we have the simplified expression of 1 - cos 44, we substitute it back into the original expression: (1/(1 - cos 44))^2.
Step 4: Simplify the fraction.
To simplify the fraction, we need to find a common denominator for the numerator and denominator. In this case, the common denominator is (1 - cos 44)(1 - cos 44), which is the denominator squared.
The expression becomes [(1^2)/((1 - cos 44)^2)].
Step 5: Simplify the numerator and denominator.
Simplify the numerator, 1^2, which is equal to 1.
Simplify the denominator, (1 - cos 44)^2, by squaring the whole expression.
The expression simplifies to [(1)/((1 - cos 44)(1 - cos 44))].
Step 6: Simplify further, if possible.
If you want to simplify the expression further, you might need additional information. However, if you just want to simplify it algebraically, the expression is already in its simplest form.
Please note that if a numerical value for cos 44 is given, you can substitute it into the expression to obtain a numerical result.