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1+sin⁡xcos⁡2x\frac{1+\sin x}{\cos^{2}x}=

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Q. 1+sin⁡xcos⁡2x\frac{1+\sin x}{\cos^{2}x}=
  1. Apply Pythagorean Identity: Apply the Pythagorean identity sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1 to express sin⁡2(x)\sin^2(x) in terms of cos⁡2(x)\cos^2(x).\newlinesin⁡2(x)=1−cos⁡2(x)\sin^2(x) = 1 - \cos^2(x)
  2. Rewrite Expression: Rewrite the original expression by adding and subtracting sin⁡2(x)\sin^2(x) in the numerator.\newline(1+sin⁡x)/(cos⁡2(x))=(1+sin⁡x+sin⁡2(x)−sin⁡2(x))/(cos⁡2(x))(1 + \sin x)/(\cos^2(x)) = (1 + \sin x + \sin^2(x) - \sin^2(x))/(\cos^2(x))
  3. Replace with Identity: Use the Pythagorean identity from step 11 to replace 1−sin⁡2(x)1 - \sin^2(x) with cos⁡2(x)\cos^2(x). \newline1+sin⁡x+sin⁡2(x)−sin⁡2(x)cos⁡2(x)=cos⁡2(x)+sin⁡xcos⁡2(x)\frac{1 + \sin x + \sin^2(x) - \sin^2(x)}{\cos^2(x)} = \frac{\cos^2(x) + \sin x}{\cos^2(x)}
  4. Split Fraction: Split the fraction into two parts.\newline(cos⁡2(x)+sin⁡x)/(cos⁡2(x))=cos⁡2(x)/cos⁡2(x)+sin⁡x/cos⁡2(x)(\cos^2(x) + \sin x)/(\cos^2(x)) = \cos^2(x)/\cos^2(x) + \sin x/\cos^2(x)
  5. Simplify First Part: Simplify the first part of the fraction. cos⁡2(x)cos⁡2(x)=1\frac{\cos^2(x)}{\cos^2(x)} = 1
  6. Recognize Trig Identity: Recognize the second part of the fraction as a trigonometric identity. sin⁡x/cos⁡2(x)\sin x/\cos^2(x) is the same as sin⁡x/(cos⁡x⋅cos⁡x)\sin x/(\cos x \cdot \cos x), which is tan⁡x/cos⁡x\tan x/\cos x.
  7. Combine Simplified Parts: Combine the simplified parts. 1+tan⁡xcos⁡x1 + \frac{\tan x}{\cos x}
  8. Final Simplified Form: Recognize that tan⁡x/cos⁡x\tan x/\cos x can be written as sin⁡x/cos⁡2(x)\sin x/\cos^2(x), which is already in the original expression.\newlineTherefore, the final simplified form is 1+tan⁡x/cos⁡x1 + \tan x/\cos x.

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