Prove the trigonometric identity $\sec\left(x\right)^2+\csc\left(x\right)^2=\frac{1}{\sin\left(x\right)^2\cos\left(x\right)^2}$

Step-by-step Solution

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Final answer to the problem

true

Step-by-step Solution

How should I solve this problem?

  • Prove from LHS (left-hand side)
  • Prove from RHS (right-hand side)
  • Express everything into Sine and Cosine
  • Exact Differential Equation
  • Linear Differential Equation
  • Separable Differential Equation
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
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1

Starting from the left-hand side (LHS) of the identity

$\sec\left(x\right)^2+\csc\left(x\right)^2$
2

Applying the secant identity: $\displaystyle\sec\left(\theta\right)=\frac{1}{\cos\left(\theta\right)}$

$\frac{1}{\cos\left(x\right)^2}+\csc\left(x\right)^2$
3

Applying the cosecant identity: $\displaystyle\csc\left(\theta\right)=\frac{1}{\sin\left(\theta\right)}$

$\frac{1}{\cos\left(x\right)^2}+\frac{1}{\sin\left(x\right)^2}$
4

The least common multiple (LCM) of a sum of algebraic fractions consists of the product of the common factors with the greatest exponent, and the uncommon factors

$L.C.M.=\cos\left(x\right)^2\sin\left(x\right)^2$
5

Obtained the least common multiple (LCM), we place it as the denominator of each fraction, and in the numerator of each fraction we add the factors that we need to complete

$\frac{\sin\left(x\right)^2}{\cos\left(x\right)^2\sin\left(x\right)^2}+\frac{\cos\left(x\right)^2}{\cos\left(x\right)^2\sin\left(x\right)^2}$
6

Combine and simplify all terms in the same fraction with common denominator $\cos\left(x\right)^2\sin\left(x\right)^2$

$\frac{1}{\cos\left(x\right)^2\sin\left(x\right)^2}$
7

Since we have reached the expression of our goal, we have proven the identity

true

Final answer to the problem

true

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Main Topic: Differential Calculus

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus.

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