Trigonometrical Identities — Case-Study Based 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° :

$A = \text{sin}^2 x = \text{sin}^2 30° = \Big(\dfrac{1}{2}\Big)^2 = \dfrac{1}{4} \\[1em] B = 1 - \text{cos}^2 x = 1 - \text{cos}^2 30° = 1 - \Big(\dfrac{\sqrt{3}}{2}\Big)^2 = 1 - \dfrac{3}{4} = \dfrac{1}{4}$

That is, A = B

⇒ sin² x = 1 - cos² x is an identity.

Similarly, for angle x = 60° :

(i) Find the value of tan x + cot x (say, A)

(ii) Find the value of $\dfrac{1}{\text{sin x cos x}}$ (say, B)

(iii) Write the relation between A and B.

(iv) Write the corresponding identity.

Answer

(i) Solving, for x = 60° :

$\Rightarrow \text{tan x + cot x} \\[1em] \Rightarrow \text{tan 60° + cot 60°} \\[1em] \Rightarrow \sqrt{3} + \dfrac{1}{\sqrt{3}} \\[1em] \Rightarrow \dfrac{3 + 1}{\sqrt{3}} \\[1em] \Rightarrow \dfrac{4}{\sqrt{3}} \\[1em] \Rightarrow \dfrac{4 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} \\[1em] \Rightarrow \dfrac{4\sqrt{3}}{3}.$

Hence, A = tan x + cot x = $\dfrac{4\sqrt{3}}{3}$.

(ii) Solving, for x = 60° :

$\Rightarrow \dfrac{1}{\text{sin x cos x}} \\[1em] \Rightarrow \dfrac{1}{\text{sin 60° cos 60°}} \\[1em] \Rightarrow \dfrac{1}{\dfrac{\sqrt{3}}{2} \times \dfrac{1}{2}} \\[1em] \Rightarrow \dfrac{1}{\dfrac{\sqrt{3}}{4}} \\[1em] \Rightarrow \dfrac{4}{\sqrt{3}} \\[1em] \Rightarrow \dfrac{4 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} \\[1em] \Rightarrow \dfrac{4\sqrt{3}}{3}.$

Hence, B = $\dfrac{1}{\text{sin x cos x}} = \dfrac{4\sqrt{3}}{3}$.

(iii) From parts (i) and (ii), A = $\dfrac{4\sqrt{3}}{3}$ and B = $\dfrac{4\sqrt{3}}{3}$.

∴ A = B.

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

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

Hence, the corresponding identity is tan x + cot x = $\dfrac{1}{\text{sin x cos x}}$.