Simplify Trigonometric Expressions
Question1: \(3-{3 sin^2x}\)
solution: \(3-{3 sin^2x}\)= \({3(1-sin^2x)}\) = \({3cos^2}\)
Question2: \({cos^3x + cos x ~sin^2x}\)
solution: \({cos^3x + cos x~ sin^2x}\) = \({cosx(cos^2x + sin^2x)}\) = cos x
Question3: 3 sin x + 7 cos x tan x
solution: sin x + 7 cos x tan x = 3 sin x + 7 sin x = 3 sinx + 7 sin x = 10 sin x
Question4: tan2x – 3 + tan x + 2
solution:tan2 x – 3 + tan x + 2 = (tanx − 1) (tanx − 2)
Question5: (cos θ+ sin θ)2 + (cos θ− sin θ)2
solution: (cos θ + sin θ)2 + (cos θ − sin θ )2 = cos2 θ+ 2 sin θ cos θ + sin2θ + cos2θ−2 sin θ cos θ + sin2θ
Question6: \(\frac{sin θ}{1+ cos θ}+ \frac{1+ cos θ}{sin θ}\)
solution: \(\frac{sin θ}{1+ cos θ}+ \frac{1+ cos θ}{sin θ}\) = \(\frac{sin^2θ + (1 + cos θ)^2}{(1+ cos θ)sin θ}\) = \(\frac{sin^2θ + 2 cos θ+ cos^2θ}{sin θ + cos θ}\)
= \(\frac{2 + 2 cos θ}{sin θ(1 + cos θ)}\) = \(\frac{2(1 + cos θ)}{sin θ(1 + cos θ)}\) = \(\frac{2}{sin θ}\) = 2 csc θ
Question7: \(\frac{1- sin^4x}{1 + sin^2x}\)
solution: \(\frac{1- sin^4x}{1 + sin^2x}\) = \(\frac{(1 - sin^2x(1 + sin^2x)}{1 + sin^2 x}\) = 1 − sin2 = cos2
Question8: 1 − sin θ cos θ tan θ
solution: 1 − sin θ cos θ tan θ= − sin θ cos θ \(\frac{sin x}{cos x}\) = 1 – sin2x = cos2x
Question9: \(\frac{sin θ(1 + cos θ)}{cos^2 θ + cos θ}\)
solution: \(\frac{sin θ(1 + cos θ)}{cos^2θ + cos θ}\) = \(\frac{sin θ(1 + cos θ)}{cos θ(cos θ + 1)}\) = \(\frac{sin θ}{cos θ}\) = tan θ
Question10: tan2θ cos2θ + cot2θ sin2θ
solution:: tan2 θcos^2θ + cot2θ sin^2θ = \(\frac{sin^2θ}{cos^2θ}.cos^2θ\) = 1