Question

6. Evaluate: tan53°tan37°+tan25°cot25°+cot45°+cot30°cot60°

Answer 1

Pre-requisites:

The following are the formulas for complementary angles of trigonometric ratios:

(i) sin (90° -  θ) = cos θ

(ii) cos (90° -  θ) = sin θ

(iii) tan (90° -  θ) = cot θ

(iv) cot (90° -  θ) = tan θ  

(v) sec (90° -  θ) = cosec θ

(vi) cosec (90° -  θ)  = sec θ

• Complementary Angles: Two angles are said to be complementary if their sum is 90°. Thus θ and (90° - θ) are complementary angles.

Solution:

tan53°. tan37° + tan25°. cot25° + cot45° + cot30°. cot60°

=tan(90°-37°).tan37° + 1 + 1 + cot(90°-60°).cot60°

                                                     

=cot37°.tan37° + 1 + 1 + tan60°.cot60°

=1 + 1 + 1 + 1

=4

Remember: (∴tan θ*cot θ=1 & cot 45°=1)

{check the attachment for more information about the trigonometric values}