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Exercise

$\left(seca+cosa\right)\left(seca-cosa\right)=tan^2a+sen^2a$

Step-by-step Solution

1

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

$\left(\sec\left(a\right)+\cos\left(a\right)\right)\left(\sec\left(a\right)-\cos\left(a\right)\right)$
2

The sum of two terms multiplied by their difference is equal to the square of the first term minus the square of the second term. In other words: $(a+b)(a-b)=a^2-b^2$.

$\sec\left(a\right)^2-\cos\left(a\right)^2$
3

Applying the trigonometric identity: $\sec\left(\theta \right)^2 = 1+\tan\left(\theta \right)^2$

$1+\tan\left(a\right)^2-\cos\left(a\right)^2$
Why is tan(x)^2+1 = sec(x)^2 ?
4

Apply the trigonometric identity: $1-\cos\left(\theta \right)^2$$=\sin\left(\theta \right)^2$, where $x=a$

$\sin\left(a\right)^2+\tan\left(a\right)^2$
Why is 1 - cos(x)^2 = sin(x)^2 ?
5

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

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

$\frac{1-sin\left(x\right)}{cos\left(x\right)}=\frac{cos\left(x\right)}{1+sin\left(x\right)}$

$\frac{1}{sec\left(x\right)+tan\left(x\right)}=sec\left(x\right)-tan\left(x\right)$

$\frac{1-\sin\:\left(x\right)}{\cos\:\left(x\right)}=\frac{\cos\:\left(x\right)}{1+\sin\:\left(x\right)}$

$\left(1+sinx\right)\left(1-sinx\right)=cos^2x$

$\sec\:\left(x\right)-\tan\:\left(x\right)=\frac{\cos\:\left(x\right)}{1+\sin\:\left(x\right)}$

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

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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