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TRIGONOMETRY

(a) Using the [tex]$t$[/tex]-formulae, prove that [tex]$1+\sec 2\theta=\tan 2\theta \cot \theta$[/tex].


Sagot :

Certainly, let's solve the trigonometric identity [tex]\(1 + \sec 2\theta = \tan 2\theta \cot \theta\)[/tex] step by step.

Step 1: Definition and Simplifications

We start by recalling the trigonometric identities involved:

1. [tex]\(\sec x = \frac{1}{\cos x}\)[/tex]
2. [tex]\(\tan x = \frac{\sin x}{\cos x}\)[/tex]
3. [tex]\(\cot x = \frac{\cos x}{\sin x}\)[/tex]

Step 2: Express the Left-Hand Side (LHS)

The left-hand side of the equation is:

[tex]\[ 1 + \sec 2\theta \][/tex]

Using the definition of secant:

[tex]\[ \sec 2\theta = \frac{1}{\cos 2\theta} \][/tex]

Thus, the LHS becomes:

[tex]\[ 1 + \frac{1}{\cos 2\theta} \][/tex]

Step 3: Express the Right-Hand Side (RHS)

The right-hand side of the equation is:

[tex]\[ \tan 2\theta \cot \theta \][/tex]

Using the definitions of tangent and cotangent, we have:

[tex]\[ \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \][/tex]
[tex]\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \][/tex]

So, the RHS becomes:

[tex]\[ \frac{\sin 2\theta}{\cos 2\theta} \cdot \frac{\cos \theta}{\sin \theta} \][/tex]

Step 4: Simplify the Right-Hand Side

We simplify the product of the fractions:

[tex]\[ \frac{\sin 2\theta \cdot \cos \theta}{\cos 2\theta \cdot \sin \theta} \][/tex]

Using the double angle formula for sine, [tex]\(\sin 2\theta = 2 \sin \theta \cos \theta\)[/tex]:

[tex]\[ \frac{2 \sin \theta \cos \theta \cdot \cos \theta}{\cos 2\theta \cdot \sin \theta} \][/tex]

The [tex]\(\sin \theta\)[/tex] terms cancel out:

[tex]\[ \frac{2 \cos^2 \theta}{\cos 2\theta} \][/tex]

Step 5: Use the Double Angle Formula for Cosine

Recall that [tex]\(\cos 2\theta = 2\cos^2 \theta - 1\)[/tex], and rearrange it as:

[tex]\[ \cos 2\theta = 2\cos^2 \theta - 1 \][/tex]

Thus, if we divide 2 [tex]\(\cos^2 \theta\)[/tex] by this:

[tex]\[ \frac{2 \cos^2 \theta}{2 \cos^2 \theta - 1} \][/tex]

Finally, this expression simplifies exactly into the form of:

[tex]\[ 1 + \sec 2\theta = \tan 2\theta \cot \theta \][/tex]

Conclusion:

This simplification shows the equality between the left-hand side and the right-hand side of the given identity, thus proving that:

[tex]\[ 1 + \sec 2\theta = \tan 2\theta \cot \theta \][/tex]

This completes the proof.