$\cot^2 \theta - \frac{1}{\sin^2 \theta}$ is equal to :

Step 1: Recall trigonometric identities

We need to simplify the given expression. To do this, we will use fundamental trigonometric identities. First, recall that the cosecant function, $\csc \theta$, is the reciprocal of the sine function, $\sin \theta$. Also, there is a Pythagorean identity relating $\cot^2 \theta$ and $\csc^2 \theta$.

Step 2: Substitute the reciprocal identity

Substitute the reciprocal identity $\frac{1}{\sin^2 \theta} = \csc^2 \theta$ into the original expression. This transforms the expression into a form that can be further simplified using another identity.

Step 3: Apply the Pythagorean identity

From the Pythagorean identity $\cot^2 \theta + 1 = \csc^2 \theta$, we can substitute $\csc^2 \theta$ with $(\cot^2 \theta + 1)$ in our expression. This substitution is key to simplifying the expression to a constant value.

Step 4: Simplify the expression

Now, distribute the negative sign and combine like terms. The $\cot^2 \theta$ terms cancel each other out, leaving us with a simple numerical result.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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