To verify the identities, we will simplify both sides of each identity step by step.
Part (a)
We need to verify: \[ (\sin x - \cos x)(\csc x + \sec x) = \tan x - \cot x. \]
LHS: We start with the left-hand side: \[ (\sin x - \cos x)(\csc x + \sec x). \] Recall that: \[ \csc x = \frac{1}{\sin x}, \quad \sec x = \frac{1}{\cos x}. \] Substituting these into the expression, we have: \[ = (\sin x - \cos x)\left(\frac{1}{\sin x} + \frac{1}{\cos x}\right). \] This can be rewritten as: \[ = (\sin x - \cos x)\left(\frac{\cos x + \sin x}{\sin x \cos x}\right). \] Now multiply: \[ = \frac{(\sin x - \cos x)(\sin x + \cos x)}{\sin x \cos x}. \] Next, we use the difference of squares: \[ \sin^2 x - \cos^2 x = -(\cos^2 x - \sin^2 x). \] Thus: \[ = \frac{\sin^2 x - \cos^2 x}{\sin x \cos x} = \frac{-(\cos^2 x - \sin^2 x)}{\sin x \cos x} = -\frac{\cos^2 x - \sin^2 x}{\sin x \cos x}. \] Now we can express this in terms of tangent and cotangent: \[ = -\left(\frac{\cos x}{\sin x} \cdot \frac{\cos x - \sin x}{\sin x}\right) = -\cot x + \tan x. \] So: \[ \text{LHS} = \tan x - \cot x. \]
RHS: Now we verify the right-hand side: \[ \tan x - \cot x = \frac{\sin x}{\cos x} - \frac{\cos x}{\sin x}. \] Finding a common denominator, we have: \[ = \frac{\sin^2 x - \cos^2 x}{\sin x \cos x}. \] This matches our LHS.
Therefore, we have shown: \[ (\sin x - \cos x)(\csc x + \sec x) = \tan x - \cot x. \] The identity is verified.
Part (b)
Now we need to verify: \[ \sec^2 x - \csc^2 x = \tan^2 x - \cot^2 x. \]
LHS: We start with the left-hand side: \[ \sec^2 x - \csc^2 x. \] Recall: \[ \sec^2 x = 1 + \tan^2 x \quad \text{and} \quad \csc^2 x = 1 + \cot^2 x. \] Substituting these expressions in, we get: \[ = (1 + \tan^2 x) - (1 + \cot^2 x). \] The \(1\)'s cancel out: \[ = \tan^2 x - \cot^2 x. \]
RHS: The right-hand side is: \[ \tan^2 x - \cot^2 x. \] Both sides are equal, thus: \[ \sec^2 x - \csc^2 x = \tan^2 x - \cot^2 x. \]
The identity is verified.
Conclusion
Both identities are verified:
- \((\sin x - \cos x)(\csc x + \sec x) = \tan x - \cot x\),
- \(\sec^2 x - \csc^2 x = \tan^2 x - \cot^2 x\).
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