1 + tan^2 (5) - csc^2 (85) = ?
Would this equal 0
could someone explain this to me and the correct answer?
Correct, assuming you work in degrees,
1+tan²(5°)-csc²(85°)
=1+tan²(5°)-sec²(90-85°)
=sec²(5°)-sec²(5°)
=0
To evaluate the expression 1 + tan^2 (5) - csc^2 (85), let's break it down step by step.
1. First, let's evaluate tan^2 (5). To do this, we need to find the tangent of 5 degrees and square the result. It's important to note that trigonometric functions typically work with angles in radians, not degrees.
To convert 5 degrees to radians, we can use the conversion factor: π radians = 180 degrees.
So, 5 degrees * (π/180) radians/degree = 5π/180 radians.
Now, find the tangent of 5π/180 radians using a scientific calculator or trigonometric identity. Assuming you're using degrees mode on your calculator, you need to convert 5π/180 radians back to degrees.
Using the identity tan(x) = sin(x) / cos(x), we have:
tan(5π/180 radians) = sin(5π/180 radians) / cos(5π/180 radians)
Calculate the sine and cosine of 5π/180 radians, and divide them:
sin(5π/180 radians) / cos(5π/180 radians)
The division results in the value of tan(5π/180 radians).
2. Next, evaluate csc^2 (85). Similarly, we need to find the cosecant of 85 degrees and square the result. Again, remember that trigonometric functions typically work with angles in radians, not degrees.
Convert 85 degrees to radians using the conversion factor: π radians = 180 degrees.
So, 85 degrees * (π/180) radians/degree = 85π/180 radians.
Now, find the cosecant of 85π/180 radians using a scientific calculator or trigonometric identity. Assuming you're using degrees mode on your calculator, you need to convert 85π/180 radians back to degrees.
Using the identity csc(x) = 1 / sin(x), we have:
1 / sin(85π/180 radians)
Calculate the sine of 85π/180 radians and take the reciprocal:
1 / sin(85π/180 radians)
The reciprocal results in the value of csc(85π/180 radians).
3. Now that we have the values for tan^2 (5) and csc^2 (85), we can substitute them back into the original expression:
1 + tan^2 (5) - csc^2 (85) = 1 + (value of tan^2 (5)) - (value of csc^2(85))
4. Calculate the squared values obtained from Steps 1 and 2 and then substitute them into the expression.
Once you've evaluated the squared values, substitute them into the expression 1 + tan^2 (5) - csc^2 (85) to obtain the final result.
If the final calculation equals 0, then the original expression is equal to 0. If it doesn't equal 0, then the original expression is not equal to 0.
Please note that performing the actual calculations would require a calculator or software capable of handling trigonometric functions.