Question

5. verify each of the following identities
a) (sinx -cosx)(cscx+secx) =tanx-cotx b) sec2x- csc2x= tan2x- cot2x

Answer 1

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:
1. \((\sin x - \cos x)(\csc x + \sec x) = \tan x - \cot x\),
2. \(\sec^2 x - \csc^2 x = \tan^2 x - \cot^2 x\).