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Prove the following. prove that sec2θ + cosec2θ = sec2θ ´ cosec2θ

Chapter 6 – Trigonometry – Text Book Solution

Problem set 6| Q 5.3| Page 138

Prove the following.
(3) prove that sec²θ + cosec²θ = sec²θ ´ cosec²θ

We are given the expression to prove: $$\sec^2\theta + \csc^2\theta = \sec^2\theta \times \csc^2\theta$$ We can start with the left-hand side (LHS): \begin{align*} \sec^2\theta + \csc^2\theta &= \frac{1}{\cos^2\theta} + \frac{1}{\sin^2\theta} \quad (\text{using the definitions of secant and cosecant})\ &= \frac{\sin^2\theta + \cos^2\theta}{\sin^2\theta \cos^2\theta} \quad (\text{adding fractions and using the identity } \sin^2\theta + \cos^2\theta = 1)\ &= \frac{1}{\sin^2\theta \cos^2\theta} \quad (\text{simplifying})\ &= \frac{1}{\sin^2\theta} \times \frac{1}{\cos^2\theta} \quad (\text{rearranging terms and using the definition of cosecant and secant})\ &= \csc^2\theta \times \sec^2\theta \quad (\text{using the definition of cosecant and secant}) \end{align*} Therefore, the left-hand side (LHS) is equal to the right-hand side (RHS), so we have proved that: $$\sec^2\theta + \csc^2\theta = \sec^2\theta \times \csc^2\theta$$

Solution

We can start with the left-hand side (LHS):

sec²θ + cosec²θ

= 1/cos²θ + 1/sin²θ // using the definitions of secant and cosecant

= (sin²θ + cos²θ) / (sin²θ cos²θ) // adding fractions and using the identity sin²θ + cos²θ = 1

= 1 / (sin²θ cos²θ) // simplifying

= (1/sin²θ) x (1/cos²θ) // rearranging terms and using the definition of cosecant and secant

= cosec²θ x sec²θ // using the definition of cosecant and secant

Therefore, the left-hand side (LHS) is eθual to the right-hand side (RHS), so we have proved that:

sec²θ + cosec²θ = sec²θ x cosec²θ.

Explanation:-

We are given the expression to prove:

sec²θ + cosec²θ = sec²θ x cosec²θ

We can start with the left-hand side (LHS):

sec²θ + cosec²θ

= 1/cos²θ + 1/sin²θ (using the definitions of secant and cosecant)

= (sin²θ + cos²θ) / (sin²θ cos²θ) (adding fractions and using the identity sin²θ + cos²θ = 1)

= 1 / (sin²θ cos²θ) (simplifying)

= (1/sin²θ) x (1/cos²θ) (rearranging terms and using the definition of cosecant and secant)

= cosec²θ x sec²θ (using the definition of cosecant and secant)

Therefore, the left-hand side (LHS) is equal to the right-hand side (RHS), so we have proved that:

sec²θ + cosec²θ = sec²θ x cosec²θ.

Chapter 6 – Trigonometry – Text Book Solution

Problem Set 6 |Q 5.3| P 138

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