The given function is ${\left(\mathit{c}\mathit{o}\mathit{s}\mathit{x}\right)}^{\mathit{y}}$ = ${\left(\mathit{c}\mathit{o}\mathit{s}\mathit{y}\right)}^{\mathit{x}}$
Taking logarithm on both the sides, we obtain
ylog cosx = xlog cosy
Differentiating both sides, we obtain
log cosx × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + y × $\frac{\mathit{d}}{\mathit{d}\mathit{x}}$ (log cos x) = log cos y × $\frac{\mathit{d}}{\mathit{d}\mathit{x}}$ (x) + x × $\frac{\mathit{d}}{\mathit{d}\mathit{x}}$ (log cos y)
⇒ log cos x × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + y × $\frac{1}{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$ × $\frac{\mathit{d}}{\mathit{d}\mathit{x}}$ (cos x) = log cos y × 1 + x × $\frac{1}{\mathit{c}\mathit{o}\mathit{s}\mathit{y}}$ × $\frac{\mathit{d}}{\mathit{d}\mathit{x}}$ (cos y)
⇒ log cos x × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + $\frac{\mathit{y}}{\mathit{c}\mathit{o}\mathit{s}\mathit{x}}$ (- sin x) = log cos y + $\frac{\mathit{x}}{\mathit{c}\mathit{o}\mathit{s}\mathit{y}}$ × (- sin y) × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ log cos x × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ - y tan x - log cos y - x tan y × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ log cos x × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ + x tan y × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = log cos y + y tan x
⇒ (log cos x + x tan y) × $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = log cos y + y tan x
∴ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{\mathit{l}\mathit{o}\mathit{g}\mathit{c}\mathit{o}\mathit{s}\mathit{y}+\mathit{y}\mathit{t}\mathit{a}\mathit{n}\mathit{x}}{\mathit{l}\mathit{o}\mathit{g}\mathit{c}\mathit{o}\mathit{s}\mathit{x}+\mathit{x}\mathit{t}\mathit{a}\mathit{n}\mathit{y}}$
OR
We have,
siny = x sin (a + y)
⇒ x = $\frac{\mathit{s}\mathit{i}\mathit{n}\mathit{y}}{\mathit{s}\mathit{i}\mathit{n}\left(\mathit{a}+\mathit{y}\right)}$
Differentiating the above function we have,
1 =
⇒ $\mathit{s}\mathit{i}{\mathit{n}}^2$ (a + y) = [sin (a + y) × cos y - sin y cos (a + y)] $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ = $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ $\frac{\mathit{s}\mathit{i}{\mathit{n}}^2\left(\mathit{a}+\mathit{y}\right)}{\mathit{s}\mathit{i}\mathit{n}\left(\mathit{a}+\mathit{y}-\mathit{y}\right)}$ = $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ $\frac{\mathit{s}\mathit{i}{\mathit{n}}^2\left(\mathit{a}+\mathit{y}\right)}{\mathit{s}\mathit{i}\mathit{n}\mathit{a}}$ = $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$
⇒ $\frac{\mathit{d}\mathit{y}}{\mathit{d}\mathit{x}}$ = $\frac{\mathit{s}\mathit{i}{\mathit{n}}^2\left(\mathit{a}+\mathit{y}\right)}{\mathit{s}\mathit{i}\mathit{n}\mathit{a}}$