Question

if tan square theta + cot square theta is equal to 10 by 3 then letters determine the value of tan theta + cot theta and tan theta minus cot theta and from this letter write the value of tan theta​

Answer 1

Answer:

tanθ = 1/√3

Step-by-step explanation:

given, tan²θ+cot²θ = 10/3

for conveniency, let us take tan²θ = x

so, given, x+ (1/x) = 10/3

=> (x²+1)/x = 10/3

=> 3(x²+1) = 10(x)

=> 3x²-10x+3 = 0

=> 3x²-x -9x+3 = 0

=> x(3x-1) -3(3x-1) = 0

=> (3x-1)(x-3) = 0

=> 3x-1 = 0  or x-3=0

so, x=1/3 or x=3

ie., tan²θ = 1/3 or tan²θ=3

=> tanθ = 1/√3  or tanθ= √3

so, if tanθ = 1/√3 then cotθ=√3  and vice versa

=> tanθ+cotθ = 1/√3+√3

                      = (1+√3²)/√3

                      = (1+3)/√3

                      = 4/√3

and tanθ-cotθ = 1/√3-√3

                      = (1-√3²)/√3

                      = (1-3)/√3

                      = -2/√3

consider ,  (tanθ+cotθ)+(tanθ-cotθ) = (4/√3)+(-2/√3)

=> 2tanθ = (4-2)/√3

=> 2tanθ = 2/√3

=> tanθ = 1/√3

=> θ = 30°