If tan ( pi/4+theta ) + tan( pi/4-theta )= 3, then the value of tan^2 (pi/4+theta)+ tan^2(pi/4-theta)
Answers
Given : tan ( π/4 + θ ) + tan ( π/4- θ ) = 3
To Find : tan² ( π/4 + θ ) + tan²( π/4- θ ) = 3
Solution:
tan ( π/4 + θ ) + tan ( π/4- θ ) = 3
Squaring both sides
=> (tan ( π/4 + θ ) + tan ( π/4- θ ) )²= 3²
Using identity ( a + b)² = a² + b² + 2ab
=> tan² ( π/4 + θ ) + tan²( π/4- θ ) + 2 tan ( π/4 + θ ) tan ( π/4- θ ) = 9
tan ( θ) = cot ( π/2- θ)
=> tan ( π/4 + θ ) = cot ( π/2- (π/4 + θ)) = cot ( π/4- θ )
tanx = 1/cotx
=> tanx . cotx = 1
=> tan ( π/4 + θ ) tan ( π/4- θ ) = cot ( π/4- θ ) ) tan ( π/4- θ ) = 1
tan² ( π/4 + θ ) + tan²( π/4- θ ) + 2 tan ( π/4 + θ ) tan ( π/4- θ ) = 9
=> tan² ( π/4 + θ ) + tan²( π/4- θ ) + 2 * 1 = 9
=> tan² ( π/4 + θ ) + tan²( π/4- θ ) = 7
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