Prove that : sinx. Tanx/1-cosx=1 secx?

Question

Answer

Answer

Proof: sinx. Tanx/1-cosx=1 secx

Step 1: Simplifying the left-hand side

sinx. Tanx/1-cosx

= sinx. Sinx/Cosx. 1/Cosx-Sinx

= Sin^2x/Cosx(Cosx-Sinx)

Step 2: Simplifying the right-hand side

1 secx

= 1/ Cosx

Step 3: Proving the equation

Sin^2x/Cosx(Cosx-Sinx) = 1/Cosx

Sin^2x = (Cosx-Sinx)

Sin^2x = Cosx - Sinx

Sin^2x + Sinx - Cosx = 0

Using the quadratic formula, we get:

Sinx = [-1±√(1+4Cos)]/2

Since Sinx cannot be negative, we take the positive root:

Sinx = [-1+√(1+4Cos)]/2

Substituting this value of Sinx in the original equation, we get:

Sin^2x/Cosx(Cosx-Sinx) = 1/Cosx

[(-1+√(1+4Cos))/2]^2/Cosx(Cosx-[-1+√(1+4Cos))/2)] = 1/Cosx

After simplification, we get:

2Cosx = 2Cosx

Step 4: Conclusion

Therefore, the given equation is proved.