Question

In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B produce the same (equal) value for all values of the variable within a certain domain.

Infact, identity in mathematics is an equation that is true for all values of its variable. 

For angle x = 30°:

\[A = \sin^2 x = \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\]

\[B = 1 - \cos^2 x = 1 - \cos^2 30^\circ\]

\[{} = 1 - \left(\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{3}{4} = \frac{1}{4}\]

That is, A = B

⇒ sin² x = 1 − cos² x is an identity

Similarly, for angle x = 60°: 

1. Find the value of tan x + cot x (say, A)
2. Find the value of \[\frac{1}{\sin x \cos x}\] (say B)
3. Write the relation between A and B.
4. Write the corresponding identity.

Answer 1

(i) Solving, for x = 60°:

\[\Rightarrow \tan x + \cot x\]

\[\Rightarrow \tan 60^\circ + \cot 60^\circ\]

\[\Rightarrow \sqrt{3} + \frac{1}{\sqrt{3}}\]

\[\Rightarrow \frac{3 + 1}{\sqrt{3}}\]

\[\Rightarrow \frac{4}{\sqrt{3}}\]

\[\Rightarrow \frac{4 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}\]

\[\Rightarrow \frac{4\sqrt{3}}{3}\]

Hence, \[A = \tan x + \cot x = \frac{4\sqrt{3}}{3}\]

(ii) Solving, for x = 60°:

\[\Rightarrow \frac{1}{\sin x \cos x}\]

\[\Rightarrow \frac{1}{\sin 60^\circ \cos 60^\circ}\]

\[\Rightarrow \frac{1}{\frac{\sqrt{3}}{2} \times \frac{1}{2}}\]

\[\Rightarrow \frac{1}{\frac{\sqrt{3}}{4}}\]

\[\Rightarrow \frac{4}{\sqrt{3}}\]

\[\Rightarrow \frac{4 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}}\]

\[\Rightarrow \frac{4\sqrt{3}}{3}\]

Hence, \[B = \frac{1}{\sin x \cos x} = \frac{4\sqrt{3}}{3}\]

(iii) From parts (i) and (ii), \[A = \frac{4\sqrt{3}}{3}\] and \[B = \frac{4\sqrt{3}}{3}\]

∴ A = B

Hence, the relation between A and B is A = B.

(iv) Since A = B, that is \[\tan x + \cot x = \frac{1}{\sin x \cos x}\] for all values of x in the domain.

Hence, the corresponding identity is \[\tan x + \cot x = \frac{1}{\sin x \cos x}.\]