Question

General solution of equation 1 + cot theta = Cosec theta is-​

Answer 1

Concept:

Trigonometry is a discipline of mathematics that investigates the relationship between triangle side lengths and angles.

Given:

$1+cotx=cosecx$

To find:

The general solution

Solution:

In this question, a trigonometric equation is given to us and we have to determine the general solution of that equation.

So first, we will convert the equation in terms of sinx and cosx.

$cosec x=1+cotx\\\frac{1}{sinx}=1 + \frac{cosx}{sinx} \\\frac{1}{sinx}-\frac{cosx}{sinx} =1\\1-\sqrt{1sin^{2} x}=sinx\\1-sinx=\sqrt1-sin^{2} {x}$

Now, we will do squaring on the both the sides

$1+sin^{2} x-2sinx=1-sin^{2}x\\ 2sin^{2}x-2sinx=0\\ sin^{2}x-sinx=0\\ sinx(sinx-1)=0\\sinx=0,1$

So, after squaring on both Lhs and Rhs we got two values of x, 0 and 1

Now, since x is positive we will solve t further by applying the formula of the general solution of sinx.

$x=n\pi ,\\sinx=1\\sinx=\frac{\pi }{2} \\x=n\pi +(-1)^{n} \frac{\pi }{2}$

n⊆z

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