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:
$$sec^2theta + csc^2theta = sec^2theta times csc^2theta$$
We can start with the left-hand side (LHS):
begin{align*}
sec^2theta + csc^2theta &= frac{1}{cos^2theta} + frac{1}{sin^2theta} quad (text{using the definitions of secant and cosecant})
&= frac{sin^2theta + cos^2theta}{sin^2theta cos^2theta} quad (text{adding fractions and using the identity } sin^2theta + cos^2theta = 1)
&= frac{1}{sin^2theta cos^2theta} quad (text{simplifying})
&= frac{1}{sin^2theta} times frac{1}{cos^2theta} quad (text{rearranging terms and using the definition of cosecant and secant})
&= csc^2theta times sec^2theta 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:
$$sec^2theta + csc^2theta = sec^2theta times csc^2theta$$
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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