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Exercise

cot⁡(x)⋅sec⁡(x)=csc⁡(x)\cot\left(x\right)\cdot\sec\left(x\right)=\csc\left(x\right)

Step-by-step Solution

1

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

cot⁡(x)sec⁡(x)\cot\left(x\right)\sec\left(x\right)
2

Applying the trigonometric identity: cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot\left(\theta \right) = \frac{\cos\left(\theta \right)}{\sin\left(\theta \right)}

cos⁡(x)sin⁡(x)sec⁡(x)\frac{\cos\left(x\right)}{\sin\left(x\right)}\sec\left(x\right)
Why does cot(x) = cos(x)/sin(x) ?
3

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

cos⁡(x)sin⁡(x)1cos⁡(x)\frac{\cos\left(x\right)}{\sin\left(x\right)}\frac{1}{\cos\left(x\right)}
4

Multiplying fractions cos⁡(x)sin⁡(x)×1cos⁡(x)\frac{\cos\left(x\right)}{\sin\left(x\right)} \times \frac{1}{\cos\left(x\right)}

cos⁡(x)sin⁡(x)cos⁡(x)\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)}
5

Simplify the fraction cos⁡(x)sin⁡(x)cos⁡(x)\frac{\cos\left(x\right)}{\sin\left(x\right)\cos\left(x\right)} by cos⁡(x)\cos\left(x\right)

1sin⁡(x)\frac{1}{\sin\left(x\right)}
6

The reciprocal sine function is cosecant: 1sin⁡(x)=csc⁡(x)\frac{1}{\sin(x)}=\csc(x)

csc⁡(x)\csc\left(x\right)
7

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

true

Final answer to the exercise

true

Try other ways to solve this exercise

  • 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 Equations
  • Homogeneous Differential Equation
  • Integrate by partial fractions
  • Product of Binomials with Common Term
  • FOIL Method
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Similar Questions Solved

11−sin(x)−11+sin⁡(x)=2tan⁡(x)sec(x)\frac{1}{1-sin\left(x\right)}-\frac{1}{1+\sin\left(x\right)}=2\tan\left(x\right)sec\left(x\right)

tan(x)+cot(x)=sec(x)cosec(x)tan\left(x\right)+cot\left(x\right)=sec\left(x\right)cosec\left(x\right)

1+cos⁡2xsin⁡2x=cot⁡x\frac{1+\cos2x}{\sin2x}=\cot x

sec(x)−tan(x)sin(x)=1sec(x)sec\left(x\right)-tan\left(x\right)sin\left(x\right)=\frac{1}{sec\left(x\right)}

tan⁡(x)+cot⁡(x)=sec⁡(x)csc⁡(x)\tan\left(x\right)+\cot\left(x\right)=\sec\left(x\right)\csc\left(x\right)

tan⁡(a)+cot⁡(a)=sec⁡(a)csc⁡(a)\tan\left(a\right)+\cot\left(a\right)=\sec\left(a\right)\csc\left(a\right)

sec⁡x−sin⁡xtan⁡x=cos⁡x\sec x-\sin x\tan x=\cos x

Main Topic: Proving Trigonometric Identities

To prove a trigonometric identity, you have to show that one side of the equation can be transformed into the other side.

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cot(x)·sec(x)=csc(x)
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