If 10 sin4θ+15cos4θ=6 then the value of 16sec8θ27csc6θ+8sec6θ is:
Get the complete, step-by-step math solution for: "If 10sin^4θ + 15cos^4θ = 6 then the value of (27csc^6θ + 8sec^6θ)/(16sec^8θ) is:". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Simplify the given equation
We are given the equation 10sin4θ+15cos4θ=6. To simplify this, we can divide the entire equation by cos4θ. This will help us express the equation in terms of tanθ and secθ.
Step 2: Convert to tangent and secant
Dividing by cos4θ gives 10cos4θsin4θ+15cos4θcos4θ=cos4θ6. This simplifies to 10tan4θ+15=6sec4θ. We know that sec2θ=1+tan2θ, so sec4θ=(1+tan2θ)2. Substituting this, we get 10tan4θ+15=6(1+tan2θ)2.
Step 3: Solve for tan2θ
Expand the right side of the equation: 6(1+2tan2θ+tan4θ)=6+12tan2θ+6tan4θ. Rearranging the terms, we get 4tan4θ−12tan2θ+9=0. This is a perfect square trinomial, which can be factored as (2tan2θ−3)2=0. Solving for tan2θ, we find 2tan2θ=3, so tan2θ=23.
Step 4: Find sec2θ and csc2θ
Now that we have tan2θ=23, we can find sec2θ using the identity sec2θ=1+tan2θ. So, sec2θ=1+23=25. Similarly, we can find csc2θ using csc2θ=1+cot2θ=1+tan2θ1. This gives csc2θ=1+3/21=1+32=35.
Step 5: Substitute values into the expression
Substitute the values of csc2θ=35 and sec2θ=25 into the given expression. The expression becomes 16(25)427(35)3+8(25)3. Calculate the powers: (35)3=27125 and (25)3=8125 and (25)4=16625. Substitute these back: 16(16625)27(27125)+8(8125)=625125+125=625250. Finally, simplify the fraction 625250 by dividing both numerator and denominator by 125, which gives 52.