Question

If $A,B and C$ are the angles of a triangle, such that$\sec \left( {A - B} \right)$,$\sec A$ and$\sec \left( {A + B} \right)$are in arithmetic progression, then
A.$\cos e{c^2}A = 2\cos e{c^2}\dfrac{B}{2}$
B.$2se{c^2}A = se{c^2}\dfrac{B}{2}$
C.$2\cos e{c^2}A = \cos e{c^2}\dfrac{B}{2}$
D.$2se{c^2}B = se{c^2}\dfrac{A}{2}$

Answer 1

First, we shall analyze the given information so that we can able to solve the problem. We are given a condition that the given angles are in arithmetic progression. Generally in Mathematics, the derivative refers to the rate of change of a function with respect to a variable. First, we need to change the given equation for our convenience. To change the equation, we need to apply some suitable trigonometric identities. Then, we need to differentiate the resultant equation.
Formula to be used:
If $a,b$ and $c$ are in arithmetic progression, then $a + c = 2b$
$\sec \theta = \dfrac{1}{{\cos \theta }}$
$\cos (A + B) + \cos (A - B) = 2\cos A\cos B$
$\cos (A - B)\cos (A + B) = {\cos ^2}A{\cos ^2}B - {\sin ^2}A{\sin ^2}B$
${\sin ^2}\theta = 1 - \cos \theta$
$\left( {a + b} \right)(a - b) = {a^2} - {b^2}$
$1 + \cos \theta = 2\cos \dfrac{\theta }{2}$
$\cos \theta = \dfrac{1}{{\sec \theta }}$

Complete step by step answer:
It is given that $\sec \left( {A - B} \right),\sec A$ and $\sec \left( {A + B} \right)$ is in arithmetic progression.
Then $2\sec A = \sec \left( {A - B} \right) + \sec (A + B)$ (here we applied $a + c = 2b$ , where $a,b$ and $c$ are in A.P)
$2\sec A = \dfrac{1}{{\cos \left( {A - B} \right)}} + \dfrac{1}{{\cos (A + B)}}$ (Here we used the trigonometric identity $\sec \theta = \dfrac{1}{{\cos \theta }}$ )
$= \dfrac{{\cos (A + B) + \cos (A - B)}}{{\cos (A - B)\cos (A + B)}}$
$\Rightarrow \sec A = \dfrac{{\cos (A + B) + \cos (A - B)}}{{2\cos (A - B)\cos (A + B)}}$
$\Rightarrow \dfrac{1}{{\cos A}} = \dfrac{{\cos (A + B) + \cos (A - B)}}{{2\cos (A - B)\cos (A + B)}}$
$\Rightarrow \dfrac{1}{{\cos A}} = \dfrac{{2\cos A\cos B}}{{2\left[ {{{\cos }^2}A{{\cos }^2}B - {{\sin }^2}A{{\sin }^2}B} \right]}}$ (Here we applied the trigonometric identity $\cos (A + B) + \cos (A - B) = 2\cos A\cos B$ and $\cos (A - B)\cos (A + B) = {\cos ^2}A{\cos ^2}B - {\sin ^2}A{\sin ^2}B$ )
$\Rightarrow \dfrac{1}{{\cos A}} = \dfrac{{\cos A\cos B}}{{{{\cos }^2}A{{\cos }^2}B - {{\sin }^2}A{{\sin }^2}B}}$
$\Rightarrow {\cos ^2}A{\cos ^2}B - {\sin ^2}A{\sin ^2}B = {\cos ^2}A\cos B$
$\Rightarrow {\cos ^2}A{\cos ^2}B - \left( {1 - {{\cos }^2}B} \right)\left( {1 - {{\cos }^2}B} \right) = {\cos ^2}A\cos B$ (Here we applied ${\sin ^2}\theta = 1 - \cos \theta$ )
$\Rightarrow {\cos ^2}A{\cos ^2}B - \left[ {1 - {{\cos }^2}B - {{\cos }^2}A + {{\cos }^2}A{{\cos }^2}B} \right] = {\cos ^2}A\cos B$
$\Rightarrow {\cos ^2}A{\cos ^2}B - 1 + {\cos ^2}B + {\cos ^2}A - {\cos ^2}A{\cos ^2}B = {\cos ^2}A\cos B$
$\Rightarrow {\cos ^2}A + {\cos ^2}B - 1 = {\cos ^2}A\cos B$
$\Rightarrow {\cos ^2}A - {\cos ^2}A\cos B = 1 - {\cos ^2}B$
$\Rightarrow {\cos ^2}A\left( {1 - \cos B} \right) = 1 - {\cos ^2}B$
$\Rightarrow {\cos ^2}A(1 - \cos B) = \left( {1 - \cos B} \right)(1 + \cos B)$ (Here we used $\left( {a + b} \right)(a - b) = {a^2} - {b^2}$ )
$\Rightarrow {\cos ^2}A = 1 + \cos B$
$\Rightarrow {\cos ^2}A = 2{\cos ^2}\dfrac{B}{2}$ (In this step, we applied $1 + \cos \theta = 2\cos \dfrac{\theta }{2}$ )
$\Rightarrow \dfrac{1}{{{{\sec }^2}A}} = 2\dfrac{1}{{{{\sec }^2}\dfrac{B}{2}}}$ (In this step, we applied $\cos \theta = \dfrac{1}{{\sec \theta }}$ )
$\Rightarrow {\sec ^2}\dfrac{B}{2} = 2{\sec ^2}A$

Hence $2{\sec ^2}A = {\sec ^2}\dfrac{B}{2}$ and the option $\left( B \right)$ is the correct answer.

Note:
Since we are given that the given angles are in A.P, we need to apply the condition to start the problem. Here we have used this condition if $a,b$ and $c$ are in arithmetic progression, then $a + c = 2b$
When we are asked to find the derivation of the given equation, we need to change the given equation smartly for our convenience. Here we have applied trigonometric identities to change the equation. Then we need to analyze that where we need to apply the derivative formulae and where we need to apply the rule of differentiation while differentiating the given equation.