Krishna
1

Step 1:  Take a look at the given equations

NOTE:   \cosec \theta + \cot \theta = k ......................(1)

PROVE: \cos \theta = \frac{k^2 - 1}{k^2 + 2}

Step 2:  Search for the appropriate formula among the trigonometric identities

IDENTITY: 1 + \cot^2 \theta = \cosec^2 \theta

We can write this as

\cosec^2 \theta - \cot^2 \theta = 1

Step 3: Use the formula of (a + b)(a - b) = a^2 - b^2 to simplify

EXAMPLE: \cosec^2 \theta - \cot^2 \theta = 1

(\cosec \theta + \cot \theta)(\cosec \theta - \cot \theta) = 1

We know that \cosec \theta + \cot \theta = k

k (\cosec \theta - \cot \theta) = 1

\cosec \theta - \cot \theta = \frac{1}{k} ..........................(2)

Step 4:  Add equation (1) and (2)

EXAMPLE: \operatorname{cosec}\theta+\cot\theta+\operatorname{cosec}\theta-\cot\theta=k+\frac{1}{k}

2\cosec \theta = \frac{k^2 + 1}{k}.....................(3)

Step 4:  Subtracting the equation (2)  from (1)

EXAMPLE: \operatorname{cosec}\theta+\cot\theta-\operatorname{cosec}\theta+\cot\theta=k-\frac{1}{k}

2\cot \theta = \frac{k^2 - 1}{k}.............................(4)

Step 5: Dividing equation (4) by (3)

\frac{2\cot \theta}{2\cosec \theta} = \frac{ \frac{k^2 - 1}{k}}{ \frac{k^2 + 1}{k}}

\frac{\cos \theta}{\sin \theta} \sin \theta = \frac{k^2 - 1}{k^2 + 1}

\cos \theta = \frac{k^2 - 1}{k^2 + 1}

Hence proved