I really need help on this. Can someone show me step by step how to do this plz.
Solve the follow:
csc (tan^-1(x))
making a sketch of the situation might help
first look at tan^-1 (x/1)
construct a right-angled triangle with base angle of Ø, opposite side x, and adjacent side 1
so that tanØ = x/1
r^2 = x^2 + 1^2
r = √(x^1 + 1)
sinØ = x/√(x^1 + 1)
then cscØ = √(x^2 + 1)/x
csc (tan^-1(x)) = √(x^2 + 1)/x
Thank you
To solve csc(tan^-1(x)), we need to break it down into individual steps. Here's a step-by-step guide to solving it:
Step 1: Understand the Trigonometric Functions
To solve this expression, it's important to know the two trigonometric functions involved:
- The arctangent function (tan^-1(x)) returns an angle whose tangent is equal to x.
- The cosecant function (csc(x)) is the reciprocal of the sine function: csc(x) = 1 / sin(x).
Step 2: Rewrite the Expression
Begin by rewriting the given expression as follows:
csc(tan^-1(x)) = 1 / sin(tan^-1(x))
Step 3: Use the Pythagorean Identity
The Pythagorean Identity states that sin^2(x) + cos^2(x) = 1. By dividing both sides of the equation by sin^2(x), we get:
cos^2(x) / sin^2(x) + 1 = 1 / sin^2(x)
Using the reciprocal identity cos^2(x) / sin^2(x) = 1 / tan^2(x), we can rewrite the equation as:
1 / tan^2(x) + 1 = 1 / sin^2(x)
Step 4: Apply the Reciprocal Identity
Since 1 / sin^2(x) = csc^2(x), we can rewrite the equation as:
1 / tan^2(x) + 1 = csc^2(x)
Step 5: Substitute into the Expression
Substitute the above equation into the original expression:
1 / sin(tan^-1(x)) = 1 / tan^2(x) + 1
Step 6: Simplify the Expression
To simplify the expression further, combine the fractions by finding a common denominator:
1 / sin(tan^-1(x)) = (1 + tan^2(x)) / tan^2(x)
Step 7: Use the Reciprocal Identity
Since (1 + tan^2(x)) / tan^2(x) = csc^2(x), we obtain:
1 / sin(tan^-1(x)) = csc^2(x)
Step 8: Final Result
The final solution is:
csc(tan^-1(x)) = csc^2(x)
Therefore, csc(tan^-1(x)) simplifies to csc^2(x).