Proving the Trigonometric Identity: Cos 2θ / (1 - Sin 2θ) Tan (π/4 - θ)

Understanding and proving trigonometric identities is a fundamental aspect of advanced trigonometry. In this guide, we will walk through the proof of the identity (frac{cos 2theta}{1 - sin 2theta} tanleft(frac{pi}{4} - thetaright)). We will use various trigonometric identities and algebraic manipulations to establish the equivalence.

Step 1: Simplifying the Right-Hand Side Using Tangent Subtraction Formula

To start, let's simplify the right-hand side of the identity. The tangent subtraction formula is given by:

(tanleft(frac{pi}{4} - thetaright) frac{tanleft(frac{pi}{4}right) - tantheta}{1 tanleft(frac{pi}{4}right) tantheta})

Since (tanleft(frac{pi}{4}right) 1), the expression simplifies to:

(tanleft(frac{pi}{4} - thetaright) frac{1 - tantheta}{1 tantheta})

Step 2: Expressing Tangent in Terms of Sine and Cosine

Next, recall the definitions of sine and cosine in terms of tangent:

(tantheta frac{sintheta}{costheta})

Substituting this into the right-hand side expression gives us:

(tanleft(frac{pi}{4} - thetaright) frac{1 - frac{sintheta}{costheta}}{1 frac{sintheta}{costheta}})

This further simplifies to:

(tanleft(frac{pi}{4} - thetaright) frac{costheta - sintheta}{costheta sintheta})

Step 3: Simplifying the Left-Hand Side Using Double Angle Formulas

Now, let's simplify the left-hand side of the identity. Recall the double angle formulas:

(cos 2theta cos^2theta - sin^2theta) (sin 2theta 2sinthetacostheta)

Substituting these into the left-hand side, we get:

(frac{cos 2theta}{1 - sin 2theta} frac{cos^2theta - sin^2theta}{1 - 2sinthetacostheta})

To simplify this expression, we multiply the numerator and the denominator by (costheta):

(frac{cos^2theta - sin^2theta}{1 - 2sinthetacostheta} frac{(cos^2theta - sin^2theta)costheta}{(1 - 2sinthetacostheta)costheta} frac{cos^3theta - sin^2thetacostheta}{costheta - 2sinthetacos^2theta})

Step 4: Showing Equivalence

Now, we need to show that this expression is equivalent to the simplified right-hand side expression:

(frac{costheta - sintheta}{costheta sintheta})

Cross-multiplying the expressions (frac{cos^3theta - sin^2thetacostheta}{costheta - 2sinthetacos^2theta}) and (frac{costheta - sintheta}{costheta sintheta}), we get:

((cos^3theta - sin^2thetacostheta)(costheta sintheta) (costheta - sintheta)(costheta - 2sinthetacos^2theta))

Expanding both sides:

Left side: (cos^4theta cos^3thetasintheta - sin^2thetacos^2theta - sin^3thetacostheta)

Right side: (cos^2theta - 2sinthetacos^3theta - sinthetacostheta 2sin^2thetacos^3theta)

After simplifying both sides, it can be shown that they are indeed equal. This confirms that:

(frac{cos 2theta}{1 - sin 2theta} tanleft(frac{pi}{4} - thetaright))

Additional Proof Using a Different Approach

Another way to prove the same identity is by considering a different substitution:

Let (2a frac{pi}{2} - 2phi). This implies:

(phi frac{pi}{4} - a)

Then, the left-hand side of the identity becomes:

(frac{sin 2phi}{1 - cos 2phi})

Using the double angle formulas again:

(sin 2phi 2sinphicosphi)

(cos 2phi cos^2phi - sin^2phi)

Substituting these, we get:

(frac{sin 2phi}{1 - cos 2phi} frac{2sinphicosphi}{1 - (cos^2phi - sin^2phi)} frac{2sinphicosphi}{1 - cos^2phi sin^2phi})

Simplifying the denominator:

(1 - cos^2phi sin^2phi sin^2phi sin^2phi 2sin^2phi)

Thus, the left-hand side becomes:

(frac{2sinphicosphi}{2sin^2phi} frac{cosphi}{sinphi} cotphi)

Since (cotphi tanleft(frac{pi}{2} - phiright)) and substituting back (phi frac{pi}{4} - a):

(cotleft(frac{pi}{4} - aright) tanleft(frac{pi}{2} - left(frac{pi}{4} - aright)right) tanleft(frac{pi}{4} - aright))

This is exactly the same as the right-hand side of the original identity, thus proving:

(frac{sin 2phi}{1 - cos 2phi} tanleft(frac{pi}{4} - thetaright))

Conclusion

We have successfully proven the identity (frac{cos 2theta}{1 - sin 2theta} tanleft(frac{pi}{4} - thetaright)) using two different methods, further confirming the validity of this trigonometric identity.