Evaluate: 5 cos²60° + 4 sec²30° − tan²45°. (Give the exact value.)

67/12. cos 60° = 1/2 so cos²60° = 1/4; sec 30° = 2√3 so sec²30° = 4/3; tan 45° = 1 so tan²45° = 1. Value = 5(1/4) + 4(4/3) − 1 = 5/4 + 16/3 − 1 = 15/12 + 64/12 − 12/12 = 67/12.

Evaluate: 5 co $s^{2}$ 60° + 4 se $c^{2}$ 30° − ta $n^{2}$ 45°. (Give the exact value.)

Class 10Introduction to TrigonometryTrigonometry

A $\frac{67}{12}$
B1
C $\frac{5}{4}$
D $\frac{16}{3}$

Step-by-Step Solution

cos 60° =

\frac{1}{2}

so co

s^{2}

60° =

\frac{1}{4}

; sec 30° =

\frac{2}{3}

so se

c^{2}

30° =

\frac{4}{3}

; tan 45° = 1 so ta

n^{2}

45° = 1. Value =

5 (\frac{1}{4}) + 4 (\frac{4}{3}) - 1 = \frac{5}{4} + \frac{16}{3} - 1 = \frac{15}{12} + \frac{64}{12} - \frac{12}{12} = \frac{67}{12}

Answer: A)

\frac{67}{12}

Key idea: Standard angle values (must memorise)

| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |

**Memory pattern for sin**: 0/2, 1/2, √2/2, √3/2, √4/2 = 0, 1/2, √2/2, √3/2, 1. (Take square roots of 0,1,2,3,4 over 2.)
**For cos**: same pattern but in REVERSE order.
**For tan**: tan = sin/cos.

These values appear in every standardised assessment. Memorise the table cold.

Master this chapter in 30 minutes a day.

Don't just memorise the answer — build a daily practice habit.

Practice 5 Similar Questions Free →

All Introduction to Trigonometry practice questions

More Class 10 Introduction to Trigonometry practice

Assertion (A): sin 60° = √32. Reason (R): In an equilateral triangle of side 2, the…Simplify: cos 38° · cos 52° − sin 38° · sin 52°. (Hint: 38° and 52° are complementary.)If sin A = 1/2, what is the value of 3 cos A − 4 cos³A? (A is acute.)The maximum value that sin θ can take is:Assertion (A): tan(90° − θ) = cot θ. Reason (R): In a right triangle the two acute…

Evaluate: 5 cos260° + 4 sec230° − tan245°. (Give the exact value.)

Step-by-Step Solution

Key idea: Standard angle values (must memorise)

Master this chapter in 30 minutes a day.

More Class 10 Introduction to Trigonometry practice

Evaluate: 5 co $s^{2}$ 60° + 4 se $c^{2}$ 30° − ta $n^{2}$ 45°. (Give the exact value.)