Exploring Trigonometric Identities: Finding sin2x sec2y - cot2y with Given tanx and siny

When dealing with trigonometric identities and given specific values for trigonometric functions, we can explore the relationships between sin, cos, sec, and cot. This article delves into finding the value of sin2x sec2y - cot2y when tanx √3 and siny √2/2.

Step-by-Step Solution

Given Information: tanx √3 siny √2/2 Determine the Values of Sine, Cosine, Secant, and Cotangent: Since tanx √3, we know that tanx sinx/cosx. This implies sinx √3/2 and cosx 1/2 (since we are dealing with acute angles, all trigonometric ratios are positive). For siny √2/2, we can determine cosy √2/2 (since sin2y cos2y 1), and thus tany 1 and coty 1. Calculate sin2x sec2y - cot2y: sin2x (√3/2)2 3/4 sec2y 1/cos2y 1/(√2/2)2 1/1/2 2 cot2y 12 1

Therefore, (sin2x sec2y - cot2y) (3/4 * 2) - 1 6/4 - 1 3/2 - 1 1/2 7/4.

Alternative Approach

We can also solve the problem using the angles themselves:

x tan-1 √3 π/3 where x is in the first quadrant. y sin-1 √2/2 π/4 where y is also in the first quadrant. Using these angles, we find: sin(2x) sin(2π/3) √3/2 sec(2y) sec(π/2) 1/cos(π/4) 2 cot(2y) cot(π/2) 1/tan(π/4) 1 Therefore, sin2(2x) sec2(2y) - cot2(2y) (3/4 * 2) - 1 7/4.

Conclusion

The value of sin2x sec2y - cot2y is 7/4 when tanx √3 and siny √2/2.

Further Exploration

Understanding and applying trigonometric identities and the relationships between trigonometric functions is crucial in solving complex trigonometric problems. This exercise highlights the importance of acute angles and the specific values of trigonometric functions in the first quadrant. Beyond these specific values, the general trigonometric identities such as sin2θ cos2θ 1 and 1 tan2θ sec2θ are always useful in solving similar problems.