Question

The value of $(1-\cot {23}^{\circ })(1-\cot {22}^{\circ })=\sqrt{k}$,the value of $k$ must be equal to

Answer 1

The value of $(1-\cot {23}^{\circ })(1-\cot {22}^{\circ })$

$\Rightarrow \frac{(\sin {23}^{\circ }-\cos {23}^{\circ })(\sin {22}^{\circ }-\cos {22}^{\circ })}{\sin {23}^{\circ }\sin {22}^{\circ }}$

Multiply and divide by $\sqrt{2}$ and you can observe that we can apply $\sin (A-B)$.

$(\sin {23}^{\circ }-\cos {23}^{\circ })=\sqrt{2}.(\cos {45}^{\circ }\sin {23}^{\circ }-\sin {45}^{\circ }\cos {23}^{\circ })$

Similarly for $(\sin {22}^{\circ }-\cos {22}^{\circ })=\sqrt{2}.(\cos {45}^{\circ }\sin {22}^{\circ }-\sin {45}^{\circ }\cos {22}^{\circ })$

$\Rightarrow \frac{\sqrt{2}\sin ({23}^{\circ }-{45}^{\circ }).\sqrt{2}\sin ({22}^{\circ }-{45}^{\circ })}{\sin {23}^{\circ }\sin {22}^{\circ }}=\sqrt{4}$