What is \((1 + \tan\alpha \tan\beta)^2 + (\tan\alpha - \tan\beta)^2\) equal to ?
Simplifying the Trigonometric Expression
We are asked to find the value of the expression \((1 + \tan\alpha \tan\beta)^2 + (\tan\alpha - \tan\beta)^2\). To solve this, we will use algebraic expansion and fundamental trigonometric identities.
Step 1: Expand the Squared Terms
Let's expand each squared term separately:
- Expanding the first term, \((1 + \tan\alpha \tan\beta)^2\): Using the identity \((a+b)^2 = a^2 + 2ab + b^2\), where \(a=1\) and \(b=\tan\alpha \tan\beta\): \( (1 + \tan\alpha \tan\beta)^2 = 1^2 + 2(1)(\tan\alpha \tan\beta) + (\tan\alpha \tan\beta)^2 \) \( = 1 + 2\tan\alpha \tan\beta + \tan^2\alpha \tan^2\beta \)
- Expanding the second term, \((\tan\alpha - \tan\beta)^2\): Using the identity \((a-b)^2 = a^2 - 2ab + b^2\), where \(a=\tan\alpha\) and \(b=\tan\beta\): \( (\tan\alpha - \tan\beta)^2 = (\tan\alpha)^2 - 2(\tan\alpha)(\tan\beta) + (\tan\beta)^2 \) \( = \tan^2\alpha - 2\tan\alpha \tan\beta + \tan^2\beta \)
Step 2: Combine the Expanded Terms
Now, we add the results from Step 1:
\( (1 + 2\tan\alpha \tan\beta + \tan^2\alpha \tan^2\beta) + (\tan^2\alpha - 2\tan\alpha \tan\beta + \tan^2\beta) \)Combine like terms. Notice that the terms \(2\tan\alpha \tan\beta\) and \(-2\tan\alpha \tan\beta\) cancel each other out:
\( = 1 + \tan^2\alpha \tan^2\beta + \tan^2\alpha + \tan^2\beta \)Step 3: Rearrange and Apply Trigonometric Identities
Rearrange the terms to group them conveniently:
\( = (1 + \tan^2\alpha) + (\tan^2\beta + \tan^2\alpha \tan^2\beta) \)Factor out \(\tan^2\beta\) from the second group:
\( = (1 + \tan^2\alpha) + \tan^2\beta(1 + \tan^2\alpha) \)Now, we use the fundamental trigonometric identity: \(\sec^2 x = 1 + \tan^2 x\). Applying this identity to \((1 + \tan^2\alpha)\):
\( = \sec^2\alpha + \tan^2\beta (\sec^2\alpha) \)Factor out the common term \(\sec^2\alpha\):
\( = \sec^2\alpha (1 + \tan^2\beta) \)Apply the identity \(\sec^2 x = 1 + \tan^2 x\) again to \((1 + \tan^2\beta)\):
\( = \sec^2\alpha \sec^2\beta \)Conclusion
The expression \((1 + \tan\alpha \tan\beta)^2 + (\tan\alpha - \tan\beta)^2\) simplifies to \(\sec^2\alpha \sec^2\beta\). This matches the second option provided.
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