Exact value of cos 20 deg + 2sin^2 (55 deg) - sqrt(2) * sin 65 deg * k 8. (A) 1 (B) 1/(sqrt(2)) (C) sqrt(2) (D) zero

Key tools you must know

1. Pythagorean identity (double-angle form)

2. Co-function identity

3. Angle–sum formula for sine

4. Exact values at 45°

Student’s logical roadmap

1. Convert the squared sine term: Replace $2\sin^2 55°$ by $1-\cos110°$. Because $110° = 180°-70°$, we turn $\cos110°$ into $-\cos70°$. So the whole piece simplifies to $1+\cos70°$.

2. Expand the 65° sine: Recognise $65° = 45° + 20°$. Use the sum formula and exact $\sin45°, \cos45°$ values, then multiply by $\sqrt2$ so the pesky $\sqrt2$ cancels neatly.

3. Collect like terms: You will get a $+\cos20°$ from the first chunk and a $-\cos20°$ from the expanded chunk – they cancel. What remains is $1 + \cos70° - \sin20°$.

4. Use the co-function rule: Replace $\sin20°$ with $\cos70°$. They also cancel, leaving 1.

5. Match to the options. The only choice equal to 1 is option (A).

What’s the question?

We have to find the exact (not decimal-approximate) value of this funny looking mix of cos and sin angles:

cos 20°  +  2·sin²55°  –  √2·sin65°

How do we tame it?

1. Turn big things into small, friendly ones. We know some special angles like 45°, 60°, 30°. Notice 55° and 65° are close to 45° plus or minus 10° or 20°.
2. Use identities as Lego bricks. Identities are like Lego pieces that let you replace a messy block with a simpler, nicer-looking one that fits exactly.
   • $2\sin^2 x = 1-\cos(2x)$ (Pythagorean double-angle)
   • $\sin(45°+\theta) = \frac{\sqrt2}{2}\big(\cos\theta+\sin\theta\big)$ (sum formula)
   • $\sin(90°-\theta)=\cos\theta$ (co-function rule)
3. Watch pieces cancel out. When you substitute, terms like $+\cos20°$ and $-\cos20°$ magically disappear, leaving a very small remainder.
4. End up with a single number. All the angle stuff vanishes and we’re left with exactly 1. No calculator needed if you trust the identities!

Step-by-step computation

1. Rewrite the $2\sin^2$ term

But $\cos110° = \cos(180° - 70°) = -\cos70°$, so

2. Rewrite the $\sqrt2\,\sin65°$ term

Notice $65° = 45° + 20°$. Using the sum formula:

Because $\sin45° = \cos45° = \dfrac{\sqrt2}{2}$,

Multiply by $\sqrt2$:

3. Assemble the original expression

Original expression:

Replace each part:

4. Cancel like terms

5. Use the co-function identity ($\sin20° = \cos(90°-20°) = \cos70°$)

6. Choose the correct option

Option (A) $\boxed{1}$