Indeterminate Form - Infinity Over Infinity

Category: Differential Calculus

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Evaluate

Solution:

Substitute the value of x to the given equation, we have

Since ∞/∞ is also an indeterminate form like 0/0 and is not accepted also in Mathematics, then we have to apply the L'Hopital's Rule to the given equation. Take the derivative of the numerator and the denominator as well

Finally, substitute the value of x to the above equation

Note: Zero divided by any number and infinity (except zero) is always equal to zero.

Therefore, 

Proving Trigonometric Identities - Higher Degree

Category: Trigonometry

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Prove the trigonometric identity for  

Solution:

Consider the given equation

We have to examine first the both sides of the equation for their complexity. Since the right side of the equation is complicated, then we will simplify the right side of the equation as follows

Therefore,

Algebraic Radicals

Category: Algebra

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Perform the indicated operations and simplify

Solution:

Consider the given equation

Get the Least Common Denominator (LCD) of the two grouped terms,

Get the reciprocal of the divisor and perform the multiplication

Since the denominator contains radicals, we need to rationalize the denominator by multiplying both sides of the fraction by its conjugate term, meaning the opposite sign of the other term. 

Therefore, the answer is