Question

Consider the following equations
1 $\frac{\cos {75}^{\circ }}{\sin {15}^{\circ }}+\frac{\sin {12}^{\circ }}{\cos {78}^{\circ }}-\frac{cosec{18}^{\circ }}{\sec {72}^{\circ }}=1$

2 $\frac{\tan {50}^{\circ }+\sec {50}^{\circ }}{\cot {40}^{\circ }+cosec{40}^{\circ }}+\cos {40}^{\circ }cosec{50}^{\circ }=2$

3 $\frac{\sin {80}^{\circ }}{\cos {10}^{\circ }}-\sin {59}^{\circ }\sec {31}^{\circ }=0$

Which of these statements given above are correct?

• A. 1 only is correct
• B. 3 only is correct
• C. All 1, 2 and 3 are correct
• D. 2 and 3 are correct

Answer 1

Answer: (C)

All 1, 2 and 3 are correct

1.$\frac{\cos 75°}{\sin 15°}+\frac{\sin 12°}{\cos 78°}-\frac{\csc 18°}{\sec 72°}=1$

LHS$=\frac{\cos 75°}{\cos 75°}+\frac{\sin 12°}{\sin 12°}-\frac{\csc 18°}{\csc 18°}=1+1-1=1$

LHS=RHS

2.$\frac{\tan 50°+\sec 50°}{\cot 40°+\csc 40°}+\cos 40°\csc 50°=2$

LHS$=\frac{\tan 50°+\sec 50°}{\cot 40°+\csc 40°}+\cos 40°\csc 50°$

$=\frac{\cot 40°+\csc 40°}{\cot 40°+\csc 40°}+\frac{\cos 40°}{\sin 50°}$

$=1+\frac{\sin 50°}{\sin 50°}=1+1=2$

LHS=RHS

3.$\frac{\sin 80°}{\cos 10°}-\sin 59°\sec 31°=0$

LHS$=\frac{\sin 80°}{\cos 10°}-\sin 59°\sec 31°$

$=\frac{\cos 10°}{\cos 10°}-\frac{\sin 59°}{\cos 31°}=1-\frac{\cos 31°}{\cos 31°}=1-1$

$=0$

=RHS