Proving trigonometric identities questions with answers

Questions

1. Question 1sin²x + cos²x = 1 and the reciprocal ratios

   Prove that $\sec x - \cos x = \sin x\tan x$.

   Show the worked solution

   1. Step 1: Start from the more complicated side, the left-hand side$\text{LHS} = \sec x - \cos x$
   2. Step 2: Write sec x in terms of sin x and cos xsec x = 1 ÷ cos x$= \dfrac{1}{\cos x} - \cos x$
   3. Step 3: Write it as one fraction over cos xWrite it as one fraction over cos x$= \dfrac{1 - \cos^{2} x}{\cos x}$
   4. Step 4: Use a Pythagorean identityUse 1 − cos²x = sin²x$= \dfrac{\sin^{2} x}{\cos x}$
   5. Step 5: Split off a single ratioSplit off sin x ÷ cos x = tan x$= \sin x\tan x$
   6. Step 6: This is the right-hand side$= \text{RHS}$
2. Question 2sin²x + cos²x = 1 and the reciprocal ratios

   Prove that $(\sec x - 1)(\sec x + 1) = \tan^{2} x$.

   Show the worked solution

   1. Step 1: Start from the more complicated side, the left-hand side$\text{LHS} = (\sec x - 1)(\sec x + 1)$
   2. Step 2: Expand: difference of two squaresExpand: difference of two squares$= \sec^{2} x - 1$
   3. Step 3: Use a Pythagorean identityUse sec²x = 1 + tan²x$= \tan^{2} x$
   4. Step 4: This is the right-hand side$= \text{RHS}$
3. Question 3Double angle formulae

   Prove that $\dfrac{\cos 2x}{\cos x + \sin x} = \cos x - \sin x$.

   Show the worked solution

   1. Step 1: Start from the more complicated side, the left-hand side$\text{LHS} = \dfrac{\cos 2x}{\cos x + \sin x}$
   2. Step 2: Use a double angle formula: which form of cos 2x helps?Use cos 2x = cos²x − sin²x$= \dfrac{\cos^{2} x - \sin^{2} x}{\cos x + \sin x}$
   3. Step 3: Factorise the top: difference of two squaresFactorise the top: difference of two squares$= \dfrac{(\cos x - \sin x)(\cos x + \sin x)}{\cos x + \sin x}$
   4. Step 4: Cancel (cos x + sin x)Cancel (cos x + sin x)$= \cos x - \sin x$
   5. Step 5: This is the right-hand side$= \text{RHS}$
4. Question 4Double angle formulae

   Prove that $\cos^{4} x - \sin^{4} x = \cos 2x$.

   Show the worked solution

   1. Step 1: Start from the more complicated side, the left-hand side$\text{LHS} = \cos^{4} x - \sin^{4} x$
   2. Step 2: Difference of two squaresDifference of two squares$= (\cos^{2} x - \sin^{2} x)(\cos^{2} x + \sin^{2} x)$
   3. Step 3: Use a Pythagorean identityUse sin²x + cos²x = 1$= \cos^{2} x - \sin^{2} x$
   4. Step 4: Write it with a double angleUse cos 2x = cos²x − sin²x$= \cos 2x$
   5. Step 5: This is the right-hand side$= \text{RHS}$