Step 1: Use the formula 2tan−1x=tan−1(1−x22x).
Let x=51.
2tan−151=tan−1(1−(51)22⋅51)2tan−151=tan−1(1−25152)2tan−151=tan−1(252452)2tan−151=tan−1(52⋅2425)2tan−151=tan−1(12050)2tan−151=tan−1125
Step 2: Substitute this back into the expression and use 4π=tan−11.
The expression becomes:
tan(tan−1125−tan−11)
Step 3: Use the formula tan−1x−tan−1y=tan−1(1+xyx−y).
Let x=125 and y=1.
tan−1125−tan−11=tan−1(1+125⋅1125−1)tan−1125−tan−11=tan−1(1212+5125−12)tan−1125−tan−11=tan−1(1217−127)tan−1125−tan−11=tan−1(−177)
Step 4: Evaluate the final expression.
tan(tan−1(−177))=−177
The final answer is −177.
ii) Evaluate tan(21cos−135)
Step 1: Let θ=cos−135.
This implies cosθ=35.
We need to find tan(2θ).
Step 2: Use the half-angle formula tan2θ=sinθ1−cosθ.
First, find sinθ. Since θ=cos−135, θ is in the first quadrant, so sinθ>0.
sin2θ=1−cos2θsin2θ=1−(35)2sin2θ=1−95sin2θ=99−5sin2θ=94sinθ=94=32
Step 3: Substitute cosθ and sinθ into the half-angle formula.
tan2θ=321−35tan2θ=3233−5tan2θ=23−5
The final answer is 23−5.
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This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here are the evaluations for question 30: i) Evaluate (2 ^-1 (1)/(5) - ()/(4)) Step 1: Use the formula 2 ^-1 x = ^-1 ((2x)/(1-x^2)). Let x = (1)/(5). 2 ^-1 (1)/(5) = ^-1 ((2 · 1)/(5)1 - ((1)/(5))^2) 2 ^-1 (1)/(5) = ^-1 ((2)/(5)1 - (1)/(25)) 2 ^-1 (1)/(5) = ^-1 ((2)/(5)(24)/(25)) 2 ^-1 (1)/(5) = ^-1 ((2)/(5) · (25)/(24)) 2 ^-1 (1)/(5) = ^-1 ((50)/(120)) 2 ^-1 (1)/(5) = ^-1 (5)/(12) Step 2: Substitute this back into the expression and use ()/(4) = ^-1 1. The expression becomes: (^-1 (5)/(12) - ^-1 1) Step 3: Use the formula ^-1 x - ^-1 y = ^-1 ((x-y)/(1+xy)). Let x = (5)/(12) and y = 1. ^-1 (5)/(12) - ^-1 1 = ^-1 ((5)/(12) - 11 + (5)/(12) · 1) ^-1 (5)/(12) - ^-1 1 = ^-1 ((5-12)/(12)(12+5)/(12)) ^-1 (5)/(12) - ^-1 1 = ^-1 ((-7)/(12)(17)/(12)) ^-1 (5)/(12) - ^-1 1 = ^-1 (-(7)/(17)) Step 4: Evaluate the final expression. (^-1 (-(7)/(17))) = -(7)/(17) The final answer is -(7)/(17). ii) Evaluate ((1)/(2) ^-1 sqrt(5)3) Step 1: Let = ^-1 sqrt(5)3. This implies = sqrt(5)3. We need to find (()/(2)). Step 2: Use the half-angle formula ()/(2) = (1 - )/( ). First, find . Since = ^-1 sqrt(5)3, is in the first quadrant, so > 0. ^2 = 1 - ^2 ^2 = 1 - (sqrt(5)3)^2 ^2 = 1 - (5)/(9) ^2 = (9-5)/(9) ^2 = (4)/(9) = sqrt((4)/(9)) = (2)/(3) Step 3: Substitute and into the half-angle formula. ()/(2) = 1 - sqrt(5)3(2)/(3) ()/(2) = 3 - sqrt(5)3(2)/(3) ()/(2) = 3 - sqrt(5)2 The final answer is 3 - sqrt(5)2. Send me the next one 📸
