Question

Show that:
(1) tan 48° tan 23° tan 42°tan 67° = 1
(ii) cos 38° cos 52° -sin 38° sin 52° = 0​

Answer 1

Answer:

your answer attached in the photo

Answer 2

$\bf\underline{\underline{\orange{ Given:-:-}}}$

• tan 48° tan 23° tan 42° tan 67°
• cos 38° cos 52° – sin 38° sin 52°

$\bf\underline{\underline{\orange{ Concept:-:-}}}$

• Trigonometry and its applications

We need to know that:-

$\sf{tan \ \theta = cot(90 - \theta)}$

So, we can write tan 48 and tan 23 as:-

$\sf{tan \ 48 = cot(90 - 48) }$

$\implies \sf{cot \ 42}$

$\sf{tan \ 23 = cot(90-23)}$

$\mapsto \sf{cot \ 67}$

So, as they are complementary, they get cancelled, and hence the answer is 1.

Hence Proved!

Now, we need to know that:-

$\rm{sin \ \theta = cos(90- \theta)}$

So, we can write sin 38° sin 52° as :-

$\rm{sin \ 38 = cos(90- 38)}$

$\mapsto \rm{cos \ 52}$

$\rm{sin \ 52 = (90-52)}$

$\mapsto \rm{cos \ 38}$

cos 38° cos 52° – cos 38° cos 52° = 0

Hence Proved!