Show that: $\operatorname{cosec} 10^0-\sqrt{3} \sec 10^0=4$
Answer (1)
Rewrite cosec 1 0 ∘ and sec 1 0 ∘ in terms of sine and cosine.
Express the left-hand side (LHS) as a single fraction.
Simplify the expression using trigonometric identities, including the cosine addition formula and the identity cos ( 9 0 ∘ − x ) = sin x .
After correcting for a factor of 2, show that the LHS equals 4, which is equal to the right-hand side (RHS), thus proving the identity: 4 .
Explanation
Problem Analysis We are tasked with proving the trigonometric identity: cosec 1 0 ∘ − 3 sec 1 0 ∘ = 4
Rewriting in terms of sine and cosine Let's begin by expressing cosec 1 0 ∘ and sec 1 0 ∘ in terms of sine and cosine, respectively: cosec 1 0 ∘ = sin 1 0 ∘ 1 , sec 1 0 ∘ = cos 1 0 ∘ 1 Substituting these into the left-hand side (LHS) of the identity, we get: L H S = sin 1 0 ∘ 1 − 3 ⋅ cos 1 0 ∘ 1 = sin 1 0 ∘ cos 1 0 ∘ cos 1 0 ∘ − 3 sin 1 0 ∘
Applying Trigonometric Identities To simplify the numerator, we multiply both the numerator and the denominator by 2: L H S = 2 sin 1 0 ∘ cos 1 0 ∘ 2 ( 2 1 cos 1 0 ∘ − 2 3 sin 1 0 ∘ ) We recognize that cos 6 0 ∘ = 2 1 and sin 6 0 ∘ = 2 3 . Thus, we can rewrite the numerator using the cosine addition formula, cos ( A + B ) = cos A cos B − sin A sin B :
L H S = 2 sin 1 0 ∘ cos 1 0 ∘ 2 ( cos 6 0 ∘ cos 1 0 ∘ − sin 6 0 ∘ sin 1 0 ∘ ) = 2 sin 1 0 ∘ cos 1 0 ∘ 2 cos ( 6 0 ∘ + 1 0 ∘ ) = sin 2 0 ∘ 2 cos 7 0 ∘ Here, we used the double angle identity 2 sin x cos x = sin 2 x .
Simplifying the Expression Now, we use the identity cos ( 9 0 ∘ − x ) = sin x to rewrite cos 7 0 ∘ as sin 2 0 ∘ :
L H S = sin 2 0 ∘ 2 sin 2 0 ∘ = 2 However, we made an error in the previous step. The denominator should be sin 2 0 ∘ , so the expression becomes: L H S = sin 2 0 ∘ 2 cos 7 0 ∘ = sin 2 0 ∘ 2 sin ( 9 0 ∘ − 7 0 ∘ ) = sin 2 0 ∘ 2 sin 2 0 ∘ = 2 We need to correct for the factor of 2 we introduced in the denominator in step 3. The denominator was originally sin 1 0 ∘ cos 1 0 ∘ , which we multiplied by 2 to get sin 2 0 ∘ . So, the correct expression is: L H S = 2 1 sin 2 0 ∘ 2 cos 7 0 ∘ = sin 2 0 ∘ 4 cos 7 0 ∘ = sin 2 0 ∘ 4 sin 2 0 ∘ = 4
Conclusion Thus, we have shown that the left-hand side of the given identity simplifies to 4, which is equal to the right-hand side. Therefore, the identity is proven: cosec 1 0 ∘ − 3 sec 1 0 ∘ = 4
Examples
Trigonometric identities are fundamental in various fields such as physics, engineering, and navigation. For instance, in physics, they are used to analyze wave phenomena, such as light and sound waves. In engineering, they are crucial for designing structures and systems that involve periodic motion or oscillations. In navigation, trigonometric identities help in calculating distances and angles, especially when using GPS or other location-based technologies. By understanding and applying these identities, one can solve complex problems related to angles, distances, and periodic phenomena in a simplified manner.
